Updated On : February 13, 2024

**Reader's Digest: **Confused about CBSE Class 12 Applied Maths Algebra? Get clarity on concepts, syllabus, weightage, and the best prep tips here! 🤓📖

Recognizing the need for practical mathematics in today's fast-paced commercial world, CBSE introduced a fresh elective course - **Class 12 Applied Maths**.

And when we delve into the core of this subject, Algebra stands tall as one of its quintessential components. With Algebra crowned as one of its pivotal topics, it's imperative for scholars to gain a clear grasp.

To give you a perspective, in the CBSE Class 12 Board Exam, Algebra isn't just another chapter; it claims a whopping 10 marks out of the total 80. Hence, a focused understanding of it isn't a choice but a necessity.

Here are the points to be discussed in the blog:

**Syllabus of****CBSE Class 12 Applied Maths Algebra:**An overview of the syllabus for the year 2024, providing insights into what students can expect to learn.**Important Topics of Class 12 Applied Math Algebra:**Highlight the key topics within the algebra segment that students should focus on for better understanding and exam performance.**Table of Definitions in CBSE Class 12 Applied Maths Algebra:**Clarifying essential algebraic concepts and definitions to lay a strong foundation.**Different Types of Matrix in Algebra:**Exploring various types of matrices, such as square, rectangular, diagonal, and more, and their significance.**Transpose of a Matrix:**A detailed explanation of matrix transposition, its operation, and its importance in mathematical applications.**List of Common Errors While Solving CBSE Class 12 Applied Maths Algebra Questions:**Providing valuable tips to help students steer clear of common pitfalls when working with algebraic problems, ensuring accuracy in their solutions.

You will study various topics in the Applied Maths Algebra section: Matrices, Determinants, Cramer’s rule and its application, and Simple applications of matrices and determinants, including the Leontief input-output model for two variables.

Knowing the syllabus will help you know important Applied Maths Algebra CBSE class 12 topics and focus more on those topics to score good marks in the exam. The table below shows the CBSE class 12 Applied Maths Algebra syllabus 2024:

Topics |
Explanation of Topics |

Matrices |
Concept of matrices, Notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication, and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (restricted to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here, all matrices will have real entries). |

Determinants |
Concept of Determinant, Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors, and applications of determinants in finding the area of a triangle Adjoint and inverse of a square matrix Consistency, inconsistency, and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables (having a unique solution) using an inverse of a matrix |

Cramer’s Rule and its Application |
Concept of Cramer’s rule and its Application, Derivation of application. |

Simple applications of matrices and determinants, including Leontiff input-output model for two variables |
Concept of Simple applications of matrices and determinants, including Leontiff input-output model for two variables. |

As per the **CBSE Class 12 Applied Maths Syllabus,** the main topics that you will be studying in Algebra are:

**Matrices:**In matrices, you will study the types of matrices, zero and identity matrices, transpose of a matrix, symmetric and skew-symmetric matrices, operation on matrices, simple properties of addition, multiplication and scalar multiplication, the concept of elementary row and column operations, etc.**Determinants:**In determinants, you will be studying the determinant of a square matrix (3x3 matrices), properties of determinants, minors, cofactors, and application of determinants in finding the area of a triangle, Adjoint and inverse of a square matrix, consistency, inconsistency, and the number of solutions of the system of linear equation by examples, etc.- Cramer’s rule and its application.
- Simple applications of matrices and determinants, including the Leontief input-output model for two variables.

Here's a table with important definitions and concepts related to CBSE Class 12 Applied Mathematics Algebra:

Term/Concept | Definition |
---|---|

Algebra | A branch of mathematics that deals with symbols and the rules for manipulating those symbols. |

Polynomial | An algebraic expression consisting of variables, coefficients, and exponents combined using operations. |

Degree of a Polynomial | The highest exponent of the variable in a polynomial expression. |

Zero of a Polynomial | A value of the variable that makes the polynomial equal to zero. |

Factorization | Expressing a polynomial as a product of two or more polynomials. |

Factor Theorem | If (x - a) is a factor of a polynomial P(x), then P(a) = 0. |

Remainder Theorem | When a polynomial P(x) is divided by (x - a), the remainder is P(a). |

Synthetic Division | A shortcut method for polynomial division to find quotients and remainders. |

Rational Function | A function that can be expressed as the quotient of two polynomials. |

Partial Fractions | Breaking down a rational function into a sum of simpler fractions. |

Matrix | A rectangular array of numbers, symbols, or expressions arranged in rows and columns. |

Order of a Matrix | The number of rows and columns in a matrix. |

Scalar Multiplication | Multiplying a matrix by a single number (scalar), multiplies every element of the matrix. |

Matrix Addition and Subtraction | Combining or subtracting corresponding elements of two matrices. |

Matrix Multiplication | A defined operation where the product of two matrices is obtained by multiplying rows and columns. |

Identity Matrix | A square matrix with ones on the diagonal and zeros elsewhere. |

Transpose of a Matrix | Interchanging rows and columns of a matrix. |

Symmetric Matrix | A square matrix that is equal to its transpose. |

The inverse of a Matrix | A matrix that, when multiplied by the original matrix, gives the identity matrix. |

Cramer's Rule | A method for solving a system of linear equations using determinants. |

System of Linear Equations | A set of equations with multiple variables, all of which are linear. |

Consistent and Inconsistent Systems | A system of equations is consistent if it has at least one solution and inconsistent if it has none. |

Leontief Input-Output Model | A mathematical model used to analyze economic relationships and interdependencies between industries. |

**Read About:** **CBSE Class 12 Applied Maths Basic Financial with Mathematics**

In CBSE Class 12 Applied Mathematics Algebra, you'll encounter various matrices. Let's delve into each of these types with explanations:

**Rectangular Matrix:**A rectangular matrix is a matrix in which the number of rows is not equal to the number of columns. In other words, it has an arbitrary shape, not necessarily a square shape. For example, matrix A = 12;34;5612;34;56 is a rectangular matrix of order 3x2.**Square Matrix:**A square matrix is a matrix in which the number of rows is equal to the number of columns. These matrices have a square shape. For example, matrix B = 123;456;789123;456;789 is a square matrix of order 3x3.**Row Matrix:**A row matrix is a matrix that has exactly one row and any number of columns. It can be thought of as a horizontal list of numbers. For example, matrix D = 123123 is a row matrix of order 1x3.**Column Matrix:**A column matrix is a matrix that has exactly one column and any number of rows. It can be thought of as a vertical list of numbers. For example, matrix C = 1;2;31;2;3 is a column matrix of order 3x1.**Diagonal Matrix:**A diagonal matrix is a square matrix in which all the non-diagonal entries (those not on the main diagonal) are zero. The main diagonal is a set of elements from the top-left to the bottom-right of the matrix. For example, matrix P = 200;030;001200;030;001 and matrix Q = 400;050;006400;050;006 are diagonal matrices of order 3x3 and 2x2, respectively.**Scalar Matrix:**A scalar matrix is a special diagonal matrix where all the diagonal elements are equal. In other words, it is a diagonal matrix where all diagonal elements are the same. For example, matrix A = 40;0440;04 and matrix B = 200;020;002200;020;002 are scalar matrices of order 2x2 and 3x3, respectively.**Identity Matrix:**An identity matrix is a scalar matrix in which all the diagonal entries equal 1. It is denoted by the English alphabet 'I' or 'Iₙ' for an identity matrix of order 'n'. For example, matrices I₁, I₂, and I₃ are identity matrices of order 1x1, 2x2, and 3x3, respectively.**Zero Matrix:**A zero matrix is a matrix where all of its elements are zero. It is denoted by the English alphabet 'O' and is sometimes called a null or void matrix. For example, the O₂ and O₃ are zero matrices of different orders.**Equal Matrices:**Two matrices, A and B, are considered equal when they have the same order (i.e., the same number of rows and columns), and each element in matrix A equals the corresponding element in matrix B. This is denoted as A = B.

**Read About: CBSE Class 12 Applied Maths Probability**

The properties of scalar multiplication in algebra for matrices are as follows:

Let A and B be two matrices of the same order (m × n), and let k and p be scalars.

(i) **Scalar Multiplication Distributes Over Matrix Addition:**

- k(A + B) = kA + kB

This property means that you can distribute a scalar (k) across the sum of two matrices (A and B) by multiplying each matrix by the scalar individually.

(ii) **Scalar Addition Distributes Over Matrix Multiplication:**

- (k + p)A = kA + pA

This property states that you can distribute the sum of two scalars (k and p) across a matrix (A) by multiplying the matrix by each scalar individually.

(iii) **Scalar Multiplication Distributes Over Matrix Subtraction:**

- k(A - B) = kA - kB

Similar to property (i), you can distribute a scalar (k) across the difference of two matrices (A and B) by multiplying each matrix by the scalar individually.

**Read More: CBSE Class 12 Applied Maths Probability**

The properties of addition of matrices in algebra are as follows:

i) **Addition of Matrices with the Same Order:**

- The addition of two or more matrices is possible only when the given matrices are of the same order. The order of the resultant matrix is also the same as that of the given matrices.
- Symbolically: Am×n + Bm×n = Cm×n

ii) **Commutative Property of Matrix Addition:**

- Matrix addition is commutative, meaning that changing the order of addition does not change the result.
- Symbolically: Am×n + Bm×n = Bm×n + Am×n

iii) **Associative Property of Matrix Addition:**

- Matrix addition is associative, meaning that the grouping of matrices does not affect the result when adding three or more matrices together.
- Symbolically: Am×n + (Bm×n + Cm×n) = (Am×n + Bm×n) + Cm×n

iv) **Zero Matrix is the Additive Identity:**

- Adding the zero matrix (a matrix with all elements equal to zero) to any matrix leaves the original matrix unchanged.
- Symbolically: Am×n + Om×n = Am×n = Om×n + Am×n

v) **Additive Inverse or Negative of a Matrix:**

- The negative of a matrix is the additive inverse of the given matrix. Adding a matrix to its negative results in the zero matrix.
- Symbolically: Am×n + (-Am×n) = Om×n

**Check Out: CBSE Class 12 Applied Maths Basic Financial with Mathematics**

The properties of multiplication of matrices in algebra are as follows:

i) **Associative Property of Matrix Multiplication:**

- Matrix multiplication is associative, which means that for any three matrices A, B, and C, the order in which you multiply them does not affect the result as long as the order of multiplication is defined on both sides.
- Symbolically: (AB)C = A(BC)

ii) **Distributive Property of Matrix Multiplication over Addition and Subtraction:**

- Matrix multiplication follows the distributive property, which means that you can distribute the multiplication of a matrix A over the addition or subtraction of two other matrices B and C.
- Symbolically:
- A(B + C) = AB + AC
- A(B - C) = AB - AC

- This property holds true whenever the order of multiplication is defined on both sides.

iii) **Existence of Multiplicative Identity (Identity Matrix):**

- For a given square matrix A of order m × m, there exists a multiplicative identity matrix of the same order, denoted as Iₘ, such that AIₘ = IₘA = A.
- The identity matrix serves as a multiplicative identity, similar to how the number 1 is a multiplicative identity in regular arithmetic.

**Find Out: CBSE Class 12 Applied Maths Numerical Applications**

The properties of the adjoint (also known as the adjugate or classical adjoint) of a square matrix A of order n are as follows:

i) **Product of a Matrix and Its Adjoint:**

- The product of a matrix A and its adjoint (adj A) is equal to the determinant of A times the identity matrix of the same order (n).
- Mathematically: A(adj A) = (adj A)A = |A|Iₙ, where |A| is the determinant of A, and Iₙ is the identity matrix of order n.

ii) **Determinant of the Adjoint Matrix:**

- The determinant of the adjoint matrix (adj A) is equal to the determinant of A raised to the power of (n-1).
- Mathematically: |adj(A)| = |A|^(n-1)

The transpose of a matrix is an operation that flips the matrix over its main diagonal, which is the line from the top-left to the bottom-right corner.

This operation changes the matrix rows into columns and the columns into rows. The result is a new matrix with dimensions swapped, i.e., if the original matrix is of order m × n, the transpose will be of order n × m.

Mathematically, if you have a matrix A of order m × n, its transpose (denoted as A^T or simply A with a superscript "T") is obtained as follows:

If A = [a_ij], then A^T = [b_ij], where b_ij = a_ji.

In other words, each element in the original matrix's i-th row and j-th column becomes the transposed matrix's j-th row and i-th column element.

The properties of the transpose of a matrix are as follows:

i) **Double Transpose Property:**

- The transpose of a transpose of a matrix is the original matrix.
- Mathematically: (A^T)^T = A

ii) **Scalar Multiplication and Transpose:**

- The transpose of a scalar multiple of a matrix is equal to the scalar multiplied by the transpose of the original matrix.
- Mathematically: (kA)^T = k(A^T), where "k" is any constant.

iii) **Addition and Transpose:**

- The transpose of the sum of two matrices is equal to the sum of their transposes.
- Mathematically: (A + B)^T = A^T + B^T

iv) **Multiplication and Transpose (Reverse Order):**

- The transpose of the product of two matrices is equal to the product of their transposes taken in reverse order (i.e., transpose of B times A).
- Mathematically: (AB)^T = B^T A^T

To help you better understand the type of questions asked from the algebra topic, we have provided a few sample questions here.

**Question 1: **Under what conditions on the real numbers a, b, c, d, e, f do the simultaneous equations ax + by = e and cx + dy = f have (a) a unique solution, (b) no solution, (c) infinitely many solutions in x and y. Select values of a, b, c, d, e, f for each case, and sketch the lines ax+by = e and cx+dy = f on separate axes.

**Question 2:** For what values of a do the simultaneous equations x + 2y + a2z = 0, x + ay + z = 0, x + ay + a2z = 0 have a solution other than x = y = z = 0. For each such find the general solution of the above equations.

**Question 3: **Do 2 × 2 matrices exist satisfying the following properties? Either find such matrices or show that no such exists.

- (i) A such that A5 = I and Ai 6= I for 1 ≤ i ≤ 4
- (ii) A such that An 6= I for all positive integers n
- (iii) A and B such that AB 6= BA
- (iv) A and B such that AB is invertible and BA is singular (i.e. not invertible)
- (v) A such that A5 = I and A11 = 0

**Question 4: ** (a) Find the remainder when n2 + 4 is divided by 7 for 0 ≤ n < 7. Deduce that n2 + 4 is not divisible by 7, for every positive integer n. [Hint: write n = 7k + r where 0 ≤ r < 7.] (b) Now k is an integer such that n3 + k is not divisible by 4 for all integers n. What are the possible values of k?

**Find Out: CBSE Class 12 Applied Maths Inferential Statistics**

Certainly, here's a table outlining common errors students might encounter when solving CBSE Class 12 Applied Mathematics Algebra problems and how to avoid them:

Common Error | How to Avoid It |
---|---|

Not Paying Attention to Matrix Order |
Always check the order of matrices before performing operations like addition, subtraction, or multiplication. Ensure that the order is compatible with the operation you're performing. |

Misinterpreting Matrix Dimensions |
Be careful when interpreting matrices. Rows and columns matter. For instance, don't confuse a row matrix with a column matrix or a square matrix with a rectangular matrix. |

Overlooking Matrix Multiplication Rules |
Remember that the order of multiplication matters in matrix multiplication. (AB ≠ BA in general). Pay attention to the dimensions of the matrices to be multiplied. |

Ignoring Identity and Zero Matrices |
Understand the properties of identity and zero matrices, and use them appropriately in calculations. Know how they affect operations and equations. |

Inconsistent Use of Notation |
Maintain consistent notation throughout your work. Use the same symbols and conventions for matrices, variables, and operations. |

Incorrectly Computing Determinants |
Practice calculating determinants correctly using methods like the Sarrus Rule or cofactor expansion. Pay attention to signs and arithmetic. |

Not Checking for Equal Matrices |
When comparing matrices, ensure that they have the same order and that each element matches. Don't assume equality without verifying it. |

Misunderstanding Diagonal Matrices |
Remember that diagonal matrices have non-zero values only on the main diagonal. Ensure other entries are zero. |

Confusing Row Matrices with Scalars |
Differentiate between row matrices and scalar multiples. Don't treat a row matrix as a scalar or vice versa. |

Skipping Steps in Linear Programming |
Follow a systematic approach in linear programming problems. Clearly define objectives, constraints, and the feasible region. Don't skip key steps. |

Not Double-Checking Calculations |
Take time to double-check your calculations, especially when dealing with complex expressions or large matrices. A small arithmetic error can lead to incorrect results. |

**Explore Now: ****CBSE Class 12 Linear programming**

In CBSE Class 12 Applied Mathematics Algebra, a comprehensive understanding of matrices, determinants, and their properties is crucial for success.

These topics constitute a significant portion of the syllabus and play a vital role in various mathematical applications. To excel in this subject, students should:

**Key Takeaways:**

*Algebra is a fundamental branch of mathematics, and in CBSE Class 12 Applied Mathematics, algebra topics such as matrices, determinants, and their properties are vital.**Matrices encompass various types, including square, rectangular, row, column, diagonal, scalar, identity, and zero matrices.**Transpose of a matrix flips its rows and columns, yielding a new matrix with swapped dimensions.**Properties of matrices, such as addition, scalar multiplication, and multiplication, play a crucial role in mathematical operations.**The adjoint of a matrix has properties related to matrix products and determinants.**Understanding algebraic concepts, avoiding common errors, and practising sample questions are essential for success in CBSE Class 12 Applied Mathematics Algebra.*

**Read ****More:** **How to Apply for Rechecking in CBSE Class 12**

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February 13, 2024

**Reader's Digest: **Confused about CBSE Class 12 Applied Maths Algebra? Get clarity on concepts, syllabus, weightage, and the best prep tips here! 🤓📖

Recognizing the need for practical mathematics in today's fast-paced commercial world, CBSE introduced a fresh elective course - **Class 12 Applied Maths**.

And when we delve into the core of this subject, Algebra stands tall as one of its quintessential components. With Algebra crowned as one of its pivotal topics, it's imperative for scholars to gain a clear grasp.

To give you a perspective, in the CBSE Class 12 Board Exam, Algebra isn't just another chapter; it claims a whopping 10 marks out of the total 80. Hence, a focused understanding of it isn't a choice but a necessity.

Here are the points to be discussed in the blog:

**Syllabus of****CBSE Class 12 Applied Maths Algebra:**An overview of the syllabus for the year 2024, providing insights into what students can expect to learn.**Important Topics of Class 12 Applied Math Algebra:**Highlight the key topics within the algebra segment that students should focus on for better understanding and exam performance.**Table of Definitions in CBSE Class 12 Applied Maths Algebra:**Clarifying essential algebraic concepts and definitions to lay a strong foundation.**Different Types of Matrix in Algebra:**Exploring various types of matrices, such as square, rectangular, diagonal, and more, and their significance.**Transpose of a Matrix:**A detailed explanation of matrix transposition, its operation, and its importance in mathematical applications.**List of Common Errors While Solving CBSE Class 12 Applied Maths Algebra Questions:**Providing valuable tips to help students steer clear of common pitfalls when working with algebraic problems, ensuring accuracy in their solutions.

You will study various topics in the Applied Maths Algebra section: Matrices, Determinants, Cramer’s rule and its application, and Simple applications of matrices and determinants, including the Leontief input-output model for two variables.

Knowing the syllabus will help you know important Applied Maths Algebra CBSE class 12 topics and focus more on those topics to score good marks in the exam. The table below shows the CBSE class 12 Applied Maths Algebra syllabus 2024:

Topics |
Explanation of Topics |

Matrices |
Concept of matrices, Notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew-symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication, and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrices (restricted to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here, all matrices will have real entries). |

Determinants |
Concept of Determinant, Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors, and applications of determinants in finding the area of a triangle Adjoint and inverse of a square matrix Consistency, inconsistency, and number of solutions of a system of linear equations by examples, solving system of linear equations in two or three variables (having a unique solution) using an inverse of a matrix |

Cramer’s Rule and its Application |
Concept of Cramer’s rule and its Application, Derivation of application. |

Simple applications of matrices and determinants, including Leontiff input-output model for two variables |
Concept of Simple applications of matrices and determinants, including Leontiff input-output model for two variables. |

As per the **CBSE Class 12 Applied Maths Syllabus,** the main topics that you will be studying in Algebra are:

**Matrices:**In matrices, you will study the types of matrices, zero and identity matrices, transpose of a matrix, symmetric and skew-symmetric matrices, operation on matrices, simple properties of addition, multiplication and scalar multiplication, the concept of elementary row and column operations, etc.**Determinants:**In determinants, you will be studying the determinant of a square matrix (3x3 matrices), properties of determinants, minors, cofactors, and application of determinants in finding the area of a triangle, Adjoint and inverse of a square matrix, consistency, inconsistency, and the number of solutions of the system of linear equation by examples, etc.- Cramer’s rule and its application.
- Simple applications of matrices and determinants, including the Leontief input-output model for two variables.

Here's a table with important definitions and concepts related to CBSE Class 12 Applied Mathematics Algebra:

Term/Concept | Definition |
---|---|

Algebra | A branch of mathematics that deals with symbols and the rules for manipulating those symbols. |

Polynomial | An algebraic expression consisting of variables, coefficients, and exponents combined using operations. |

Degree of a Polynomial | The highest exponent of the variable in a polynomial expression. |

Zero of a Polynomial | A value of the variable that makes the polynomial equal to zero. |

Factorization | Expressing a polynomial as a product of two or more polynomials. |

Factor Theorem | If (x - a) is a factor of a polynomial P(x), then P(a) = 0. |

Remainder Theorem | When a polynomial P(x) is divided by (x - a), the remainder is P(a). |

Synthetic Division | A shortcut method for polynomial division to find quotients and remainders. |

Rational Function | A function that can be expressed as the quotient of two polynomials. |

Partial Fractions | Breaking down a rational function into a sum of simpler fractions. |

Matrix | A rectangular array of numbers, symbols, or expressions arranged in rows and columns. |

Order of a Matrix | The number of rows and columns in a matrix. |

Scalar Multiplication | Multiplying a matrix by a single number (scalar), multiplies every element of the matrix. |

Matrix Addition and Subtraction | Combining or subtracting corresponding elements of two matrices. |

Matrix Multiplication | A defined operation where the product of two matrices is obtained by multiplying rows and columns. |

Identity Matrix | A square matrix with ones on the diagonal and zeros elsewhere. |

Transpose of a Matrix | Interchanging rows and columns of a matrix. |

Symmetric Matrix | A square matrix that is equal to its transpose. |

The inverse of a Matrix | A matrix that, when multiplied by the original matrix, gives the identity matrix. |

Cramer's Rule | A method for solving a system of linear equations using determinants. |

System of Linear Equations | A set of equations with multiple variables, all of which are linear. |

Consistent and Inconsistent Systems | A system of equations is consistent if it has at least one solution and inconsistent if it has none. |

Leontief Input-Output Model | A mathematical model used to analyze economic relationships and interdependencies between industries. |

**Read About:** **CBSE Class 12 Applied Maths Basic Financial with Mathematics**

In CBSE Class 12 Applied Mathematics Algebra, you'll encounter various matrices. Let's delve into each of these types with explanations:

**Rectangular Matrix:**A rectangular matrix is a matrix in which the number of rows is not equal to the number of columns. In other words, it has an arbitrary shape, not necessarily a square shape. For example, matrix A = 12;34;5612;34;56 is a rectangular matrix of order 3x2.**Square Matrix:**A square matrix is a matrix in which the number of rows is equal to the number of columns. These matrices have a square shape. For example, matrix B = 123;456;789123;456;789 is a square matrix of order 3x3.**Row Matrix:**A row matrix is a matrix that has exactly one row and any number of columns. It can be thought of as a horizontal list of numbers. For example, matrix D = 123123 is a row matrix of order 1x3.**Column Matrix:**A column matrix is a matrix that has exactly one column and any number of rows. It can be thought of as a vertical list of numbers. For example, matrix C = 1;2;31;2;3 is a column matrix of order 3x1.**Diagonal Matrix:**A diagonal matrix is a square matrix in which all the non-diagonal entries (those not on the main diagonal) are zero. The main diagonal is a set of elements from the top-left to the bottom-right of the matrix. For example, matrix P = 200;030;001200;030;001 and matrix Q = 400;050;006400;050;006 are diagonal matrices of order 3x3 and 2x2, respectively.**Scalar Matrix:**A scalar matrix is a special diagonal matrix where all the diagonal elements are equal. In other words, it is a diagonal matrix where all diagonal elements are the same. For example, matrix A = 40;0440;04 and matrix B = 200;020;002200;020;002 are scalar matrices of order 2x2 and 3x3, respectively.**Identity Matrix:**An identity matrix is a scalar matrix in which all the diagonal entries equal 1. It is denoted by the English alphabet 'I' or 'Iₙ' for an identity matrix of order 'n'. For example, matrices I₁, I₂, and I₃ are identity matrices of order 1x1, 2x2, and 3x3, respectively.**Zero Matrix:**A zero matrix is a matrix where all of its elements are zero. It is denoted by the English alphabet 'O' and is sometimes called a null or void matrix. For example, the O₂ and O₃ are zero matrices of different orders.**Equal Matrices:**Two matrices, A and B, are considered equal when they have the same order (i.e., the same number of rows and columns), and each element in matrix A equals the corresponding element in matrix B. This is denoted as A = B.

**Read About: CBSE Class 12 Applied Maths Probability**

The properties of scalar multiplication in algebra for matrices are as follows:

Let A and B be two matrices of the same order (m × n), and let k and p be scalars.

(i) **Scalar Multiplication Distributes Over Matrix Addition:**

- k(A + B) = kA + kB

This property means that you can distribute a scalar (k) across the sum of two matrices (A and B) by multiplying each matrix by the scalar individually.

(ii) **Scalar Addition Distributes Over Matrix Multiplication:**

- (k + p)A = kA + pA

This property states that you can distribute the sum of two scalars (k and p) across a matrix (A) by multiplying the matrix by each scalar individually.

(iii) **Scalar Multiplication Distributes Over Matrix Subtraction:**

- k(A - B) = kA - kB

Similar to property (i), you can distribute a scalar (k) across the difference of two matrices (A and B) by multiplying each matrix by the scalar individually.

**Read More: CBSE Class 12 Applied Maths Probability**

The properties of addition of matrices in algebra are as follows:

i) **Addition of Matrices with the Same Order:**

- The addition of two or more matrices is possible only when the given matrices are of the same order. The order of the resultant matrix is also the same as that of the given matrices.
- Symbolically: Am×n + Bm×n = Cm×n

ii) **Commutative Property of Matrix Addition:**

- Matrix addition is commutative, meaning that changing the order of addition does not change the result.
- Symbolically: Am×n + Bm×n = Bm×n + Am×n

iii) **Associative Property of Matrix Addition:**

- Matrix addition is associative, meaning that the grouping of matrices does not affect the result when adding three or more matrices together.
- Symbolically: Am×n + (Bm×n + Cm×n) = (Am×n + Bm×n) + Cm×n

iv) **Zero Matrix is the Additive Identity:**

- Adding the zero matrix (a matrix with all elements equal to zero) to any matrix leaves the original matrix unchanged.
- Symbolically: Am×n + Om×n = Am×n = Om×n + Am×n

v) **Additive Inverse or Negative of a Matrix:**

- The negative of a matrix is the additive inverse of the given matrix. Adding a matrix to its negative results in the zero matrix.
- Symbolically: Am×n + (-Am×n) = Om×n

**Check Out: CBSE Class 12 Applied Maths Basic Financial with Mathematics**

The properties of multiplication of matrices in algebra are as follows:

i) **Associative Property of Matrix Multiplication:**

- Matrix multiplication is associative, which means that for any three matrices A, B, and C, the order in which you multiply them does not affect the result as long as the order of multiplication is defined on both sides.
- Symbolically: (AB)C = A(BC)

ii) **Distributive Property of Matrix Multiplication over Addition and Subtraction:**

- Matrix multiplication follows the distributive property, which means that you can distribute the multiplication of a matrix A over the addition or subtraction of two other matrices B and C.
- Symbolically:
- A(B + C) = AB + AC
- A(B - C) = AB - AC

- This property holds true whenever the order of multiplication is defined on both sides.

iii) **Existence of Multiplicative Identity (Identity Matrix):**

- For a given square matrix A of order m × m, there exists a multiplicative identity matrix of the same order, denoted as Iₘ, such that AIₘ = IₘA = A.
- The identity matrix serves as a multiplicative identity, similar to how the number 1 is a multiplicative identity in regular arithmetic.

**Find Out: CBSE Class 12 Applied Maths Numerical Applications**

The properties of the adjoint (also known as the adjugate or classical adjoint) of a square matrix A of order n are as follows:

i) **Product of a Matrix and Its Adjoint:**

- The product of a matrix A and its adjoint (adj A) is equal to the determinant of A times the identity matrix of the same order (n).
- Mathematically: A(adj A) = (adj A)A = |A|Iₙ, where |A| is the determinant of A, and Iₙ is the identity matrix of order n.

ii) **Determinant of the Adjoint Matrix:**

- The determinant of the adjoint matrix (adj A) is equal to the determinant of A raised to the power of (n-1).
- Mathematically: |adj(A)| = |A|^(n-1)

The transpose of a matrix is an operation that flips the matrix over its main diagonal, which is the line from the top-left to the bottom-right corner.

This operation changes the matrix rows into columns and the columns into rows. The result is a new matrix with dimensions swapped, i.e., if the original matrix is of order m × n, the transpose will be of order n × m.

Mathematically, if you have a matrix A of order m × n, its transpose (denoted as A^T or simply A with a superscript "T") is obtained as follows:

If A = [a_ij], then A^T = [b_ij], where b_ij = a_ji.

In other words, each element in the original matrix's i-th row and j-th column becomes the transposed matrix's j-th row and i-th column element.

The properties of the transpose of a matrix are as follows:

i) **Double Transpose Property:**

- The transpose of a transpose of a matrix is the original matrix.
- Mathematically: (A^T)^T = A

ii) **Scalar Multiplication and Transpose:**

- The transpose of a scalar multiple of a matrix is equal to the scalar multiplied by the transpose of the original matrix.
- Mathematically: (kA)^T = k(A^T), where "k" is any constant.

iii) **Addition and Transpose:**

- The transpose of the sum of two matrices is equal to the sum of their transposes.
- Mathematically: (A + B)^T = A^T + B^T

iv) **Multiplication and Transpose (Reverse Order):**

- The transpose of the product of two matrices is equal to the product of their transposes taken in reverse order (i.e., transpose of B times A).
- Mathematically: (AB)^T = B^T A^T

To help you better understand the type of questions asked from the algebra topic, we have provided a few sample questions here.

**Question 1: **Under what conditions on the real numbers a, b, c, d, e, f do the simultaneous equations ax + by = e and cx + dy = f have (a) a unique solution, (b) no solution, (c) infinitely many solutions in x and y. Select values of a, b, c, d, e, f for each case, and sketch the lines ax+by = e and cx+dy = f on separate axes.

**Question 2:** For what values of a do the simultaneous equations x + 2y + a2z = 0, x + ay + z = 0, x + ay + a2z = 0 have a solution other than x = y = z = 0. For each such find the general solution of the above equations.

**Question 3: **Do 2 × 2 matrices exist satisfying the following properties? Either find such matrices or show that no such exists.

- (i) A such that A5 = I and Ai 6= I for 1 ≤ i ≤ 4
- (ii) A such that An 6= I for all positive integers n
- (iii) A and B such that AB 6= BA
- (iv) A and B such that AB is invertible and BA is singular (i.e. not invertible)
- (v) A such that A5 = I and A11 = 0

**Question 4: ** (a) Find the remainder when n2 + 4 is divided by 7 for 0 ≤ n < 7. Deduce that n2 + 4 is not divisible by 7, for every positive integer n. [Hint: write n = 7k + r where 0 ≤ r < 7.] (b) Now k is an integer such that n3 + k is not divisible by 4 for all integers n. What are the possible values of k?

**Find Out: CBSE Class 12 Applied Maths Inferential Statistics**

Certainly, here's a table outlining common errors students might encounter when solving CBSE Class 12 Applied Mathematics Algebra problems and how to avoid them:

Common Error | How to Avoid It |
---|---|

Not Paying Attention to Matrix Order |
Always check the order of matrices before performing operations like addition, subtraction, or multiplication. Ensure that the order is compatible with the operation you're performing. |

Misinterpreting Matrix Dimensions |
Be careful when interpreting matrices. Rows and columns matter. For instance, don't confuse a row matrix with a column matrix or a square matrix with a rectangular matrix. |

Overlooking Matrix Multiplication Rules |
Remember that the order of multiplication matters in matrix multiplication. (AB ≠ BA in general). Pay attention to the dimensions of the matrices to be multiplied. |

Ignoring Identity and Zero Matrices |
Understand the properties of identity and zero matrices, and use them appropriately in calculations. Know how they affect operations and equations. |

Inconsistent Use of Notation |
Maintain consistent notation throughout your work. Use the same symbols and conventions for matrices, variables, and operations. |

Incorrectly Computing Determinants |
Practice calculating determinants correctly using methods like the Sarrus Rule or cofactor expansion. Pay attention to signs and arithmetic. |

Not Checking for Equal Matrices |
When comparing matrices, ensure that they have the same order and that each element matches. Don't assume equality without verifying it. |

Misunderstanding Diagonal Matrices |
Remember that diagonal matrices have non-zero values only on the main diagonal. Ensure other entries are zero. |

Confusing Row Matrices with Scalars |
Differentiate between row matrices and scalar multiples. Don't treat a row matrix as a scalar or vice versa. |

Skipping Steps in Linear Programming |
Follow a systematic approach in linear programming problems. Clearly define objectives, constraints, and the feasible region. Don't skip key steps. |

Not Double-Checking Calculations |
Take time to double-check your calculations, especially when dealing with complex expressions or large matrices. A small arithmetic error can lead to incorrect results. |

**Explore Now: ****CBSE Class 12 Linear programming**

In CBSE Class 12 Applied Mathematics Algebra, a comprehensive understanding of matrices, determinants, and their properties is crucial for success.

These topics constitute a significant portion of the syllabus and play a vital role in various mathematical applications. To excel in this subject, students should:

**Key Takeaways:**

*Algebra is a fundamental branch of mathematics, and in CBSE Class 12 Applied Mathematics, algebra topics such as matrices, determinants, and their properties are vital.**Matrices encompass various types, including square, rectangular, row, column, diagonal, scalar, identity, and zero matrices.**Transpose of a matrix flips its rows and columns, yielding a new matrix with swapped dimensions.**Properties of matrices, such as addition, scalar multiplication, and multiplication, play a crucial role in mathematical operations.**The adjoint of a matrix has properties related to matrix products and determinants.**Understanding algebraic concepts, avoiding common errors, and practising sample questions are essential for success in CBSE Class 12 Applied Mathematics Algebra.*

**Read ****More:** **How to Apply for Rechecking in CBSE Class 12**

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