CBSE Class 12 Mathematics Notes Chapter 3 Matrices

A matrix is a rectangular array of numbers, symbols, points, or characters, each assigned to a certain row and column. A matrix is identified by its order, which is expressed as rows and columns. The numbers, symbols, points, or characters that make up a matrix are referred to as its elements. The position of each element is determined by which row and column it belongs to. 

A variety of operations can be done on matrices, including addition, scalar multiplication, multiplication, transposition, and so on.

• Certain rules must be followed while performing certain matrix operations.
• Such as adding or subtracting only if they have the same number of rows and columns

Matrices are important for class 12 students, as well as engineering mathematics. CBSE Class 12 Mathematics Notes for Chapter 3 Matrices are given in the article below for easy preparation and understanding of the concepts involved.

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Class 12 Mathematics Chapter 3 Notes – Matrices

Matrix

Definition of Matrix

• A set of mn numbers arranged in the form of a rectangular array is called a matrix.
• A matrix of m rows and n columns is called an m x n matrix.
• An m x n matrix is usually written as

• In compact form, the above matrix is represented by A = [a_ij]_{m x n} or, A = [a_ij].
• The numbers a₁₁, a₁₂, ……etc. are known as the elements of matrix A.

Types of Matrices

The following are the different types of matrices

• Row matrix: A matrix having only one row is called a row matrix or a row vector. For example

A = [1 2 -1 -2]

• Column matrix: A matrix having only one column is called a column matrix or a column vector. For example,

Column matrix

• Square matrix: A matrix in which the number of rows is equal to the number of columns is called a square matrix. For example

Square matrix

• Diagonal matrix: A square matrix A = [a_ij]_{m x n} is called a diagonal matrix if all the elements, except those in the leading diagonal, are zero.

Diagonal matrix

• Scalar matrix: A square matrix A = [a_ij]_{m x n} is called a scalar matrix if
  • a_ij = 0, for all i ≠ j
  • a_ii = c, for all i, where c ≠ 0.

Scalar matrix

• Identity Matrix or Unit matrix: A square matrix A = [a_ij]_{m x n} is called an identity or unit matrix if
  • a_ij = 0, for all i ≠ j
  • a_ii = 1, for all i

Identity matrix

• Null or Zero matrix: A matrix whose all elements are zero is called a null matrix or a zero matrix.

Zero matrix

Equality of Matrices

• Two matrices A = [a_ij]_{m x n} and B = [b_ij]_{r x s} are equal if
  • m = r i.e. the number of rows in A equals the number of rows in B.
  • n = s i.e. the number of columns in A equals the number of columns in B.
  • a_ij = b_ij for i = 1, 2, ……., m and j = 1, 2, ……., n.
• If two matrices A and B are equal, we can write A = B, otherwise, we write A ≠ B.

Operations on Matrices

Various operations on matrices are as discussed below

Addition of Matrices

The matrix addition can be understood as - 

• Let A and B be the two matrices, each of order m x n, then their sum A + B is a matrix of order m x n.
• If A = [a_ij]_{m x n} and B = [b_ij]_{m x n} are two matrices of the same order, their sum A + B is defined to be the matrix of order m x n such that

(A + B)_ij = a_ij + b_ij for all i = 1, 2, ……., m and j = 1, 2, ……., n.

Properties of Matrix Addition

• Commutative: If A and B are two m x n matrices, then A + B = B + A. i.e. matrix addition is commutative.
• Associative: If A, B, and C are three matrices of the same order, then (A + B) + C = A + (B + C) i.e. matrix addition is associative.
• Existence of Identity: The null matrix is the identity element for matrix addition, i.e. A + 0 = 0 + A.
• Existence of Inverse: For every matrix A = [a_ij]_{m x n} there exists a matrix [- aij]m x n denoted by - A, such that A + (- A) = 0 = (- A) + A.
• Cancellation Laws: If A, B, and C are matrices of the same order, then
  • A + B = A + C ⇒ B = C
  • B + A = C + A ⇒ B = C

Multiplication of a Matrix by a Scalar

• Let A = [a_ij] be an m x n matrix and k be any number called a scalar.
• Then the matrix obtained by multiplying every element of A by k is called the scalar multiple of A by k.
• It is denoted by kA. Therefore, kA = [k a_ij]_{m x n}
• For example, if

Multiplication of a Matrix by a Scalar

Properties of Scalar Multiplication

If A = [a_ij]_{m x n} and B = [b_ij]_{m x n} are two matrices and k, l are scalars then

• k (A + B) = k A + k B
• (k + l) A = k A + l A
• (k l) A = k (l A) = l (k A)
• (- k) A = - (k A) = k (- A)
• 1 A = A
• (- 1) A = - A

Subtraction of Matrices

• For two matrices A and B of the same order, the subtraction of matrix B from matrix A is denoted by A - B and is defined as A - B = A + (-B).

Subtraction of Matrices

Multiplication of Matrices

Matrix multiplication can be understood as

• Two matrices A and B are conformable for the product AB if the number of columns in A is the same as the number of rows in B.
• If A = [a_ij]_{m x n} and B = [b_ij]_{n x p} are two matrices of order m x n and n x p respectively, then their product AB is of orders m x p.
• It is given as

Multiplication of Matrices

Properties of Matrix Multiplication

• Matrix multiplication is not commutative in general.
• Matrix multiplication is associative i.e. (AB) C = A (BC).
• Matrix multiplication is distributive over matrix addition i.e.
  • A (B + C) = AB + AC
  • (A + B)C = AC + BC whenever both sides of equality are defined.
• If A is an m x n matrix, then I_m A = A = A I_n.
• If A is an m x n matrix and O is a null matrix, then
  • A_{m x n} O_{n x p} = O_{m x p}
  • O_{p x m} A_{m x n} = O_{p x n}

Transpose of Matrix

• Let A = [a_ij] be an m x n matrix, then the transpose of matrix A, denoted by A^T or A’, is an n x m matrix such that

(A^T)_ij = a_ji for all i = 1, 2, ……., m and j = 1, 2, ……., n.

• Thus, A^T is obtained from A by changing its rows into columns and columns into rows.
• For example,

Transpose of Matrix

Properties of Transpose

• For any matrix A, (A^T)^T = A.
• For any two matrices A and B of the same order, (A + B)^T = A^T + B^T.
• If A is a matrix and k is a scalar, then (kA)^T = k (A^T).
• If A and B are two matrices such that AB is defined, then (AB)^T = B^T A^T

Symmetric and Skew-Symmetric Matrices

Symmetric Matrix

• A square matrix A = [a_ij] is called a symmetric matrix, if a_ij = a_ji for all i, j.

Symmetric Matrix

• For example,
• It follows from the definition of a symmetric matrix that A is symmetric if

a_ij = a_ji for all i, j

⇒ (A)_ji = (A^T)_ij for all i, j.

⇒ A = A^T

• Thus a square matrix is a symmetric matrix if A ^T= A.

Skew-Symmetric Matrix

• A square matrix A = [a_ij] is a skew-symmetric matrix if a_ij = - a_ji for all i, j.

Skew-Symmetric Matrix

• It follows from the definition of a skew-symmetric matrix that A is skew-symmetric if

a_ij = - a_ji for all i, j

⇒ (A)_ji = - (A^T)_ij for all i, j.

⇒ A = - A^T

⇒ A^T = - A

• Thus, a square matrix A is a skew-symmetric matrix if A^T = - A.

There are Some important List Of Top Mathematics Questions On Matrices Asked In CBSE CLASS XII

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