Class 12 — Mathematics (NCERT)

Matrices: Operations, Transpose, and Inverse

Class 12

✓By the end of this lesson students will be able to perform various operations on matrices, including addition, subtraction, scalar multiplication, and matrix multiplication.
✓By the end of this lesson students will be able to find the transpose of a matrix and understand its properties, including identifying symmetric and skew-symmetric matrices.
✓By the end of this lesson students will be able to determine if a matrix is invertible and find its inverse using both elementary operations and the adjoint method.
✓By the end of this lesson students will be able to apply the concepts of matrix operations, transpose, and inverse to solve mathematical problems.

Key concepts

Matrix Addition and Subtraction

Two matrices A and B can be added or subtracted if and only if they are of the same order. The sum or difference is obtained by adding or subtracting their corresponding elements. If A = [aᵢⱼ] and B = [bᵢⱼ] are two matrices of order m x n, then A ± B is a matrix of order m x n whose elements are given by [aᵢⱼ ± bᵢⱼ].

If A = [aᵢⱼ] and B = [bᵢⱼ] are m x n matrices, then A ± B = [aᵢⱼ ± bᵢⱼ].

Scalar Multiplication of a Matrix

If A = [aᵢⱼ] is a matrix of order m x n and k is any scalar (real number), then the scalar multiplication kA is obtained by multiplying each element of A by k. The resulting matrix kA is also of order m x n.

If A = [aᵢⱼ], then kA = [kaᵢⱼ].

Matrix Multiplication

The product of two matrices A and B, denoted by AB, is defined only if the number of columns of A is equal to the number of rows of B. If A is an m x n matrix and B is an n x p matrix, then their product AB is an m x p matrix. The element cᵢⱼ of the product matrix C = AB is obtained by multiplying the i-th row of A by the j-th column of B, i.e., by summing the products of corresponding elements. Matrix multiplication is generally not commutative (AB ≠ BA).

If A = [aᵢⱼ] (m x n) and B = [bᵢⱼ] (n x p), then C = AB = [cᵢⱼ] (m x p), where cᵢⱼ = Σ(k=1 to n) aᵢk * bkj.

Transpose of a Matrix

The transpose of a matrix A, denoted by A' or Aᵀ, is obtained by interchanging its rows and columns. If A is an m x n matrix, then its transpose A' will be an n x m matrix. The element in the i-th row and j-th column of A becomes the element in the j-th row and i-th column of A'.

If A = [aᵢⱼ] is an m x n matrix, then A' = [aⱼᵢ] is an n x m matrix.

Symmetric and Skew-Symmetric Matrices

A square matrix A is said to be a symmetric matrix if A' = A, i.e., aᵢⱼ = aⱼᵢ for all possible i and j. A square matrix A is said to be a skew-symmetric matrix if A' = -A, i.e., aᵢⱼ = -aⱼᵢ for all possible i and j. For a skew-symmetric matrix, all diagonal elements must be zero (since aᵢᵢ = -aᵢᵢ implies 2aᵢᵢ = 0, so aᵢᵢ = 0).

Symmetric: A' = A. Skew-Symmetric: A' = -A.

Elementary Operations (Row and Column Operations)

Elementary operations are a set of transformations that can be applied to rows or columns of a matrix. These operations are used to transform a matrix into a simpler form, often to find its inverse or solve systems of linear equations. There are three types of elementary operations:\n1. Interchange of any two rows (or columns): Rᵢ ↔ Rⱼ (or Cᵢ ↔ Cⱼ).\n2. Multiplication of the elements of any row (or column) by a non-zero scalar: Rᵢ → kRᵢ (or Cᵢ → kCᵢ), where k ≠ 0.\n3. Addition to the elements of any row (or column) the corresponding elements of any other row (or column) multiplied by a non-zero scalar: Rᵢ → Rᵢ + kRⱼ (or Cᵢ → Cⱼ + kCᵢ), where k ≠ 0.

Invertible Matrices

A square matrix A of order n is said to be invertible if there exists another square matrix B of the same order n such that AB = BA = I, where I is the identity matrix of order n. The matrix B is called the inverse of A and is denoted by A⁻¹. If a matrix A has an inverse, it is unique. A square matrix A is invertible if and only if it is a non-singular matrix (i.e., its determinant |A| ≠ 0).

AB = BA = I, where B = A⁻¹.

Inverse of a Matrix by Elementary Operations

To find the inverse of a square matrix A using elementary row operations, we write A = IA and apply a sequence of elementary row operations on the L.H.S. matrix A and the R.H.S. matrix I simultaneously, until the L.H.S. matrix A is transformed into the identity matrix I. The matrix obtained on the R.H.S. will then be A⁻¹. Similarly, for elementary column operations, we write A = AI and apply operations on the L.H.S. matrix A and the R.H.S. matrix I simultaneously.

Inverse of a Matrix by Adjoint Method

For a square matrix A, its inverse A⁻¹ exists if and only if A is a non-singular matrix (i.e., its determinant |A| ≠ 0). The inverse of A can be found using the formula A⁻¹ = (1/|A|) adj(A), where |A| is the determinant of A and adj(A) is the adjoint of A. The adjoint of a matrix A is the transpose of its cofactor matrix.

A⁻¹ = (1/|A|) adj(A), provided |A| ≠ 0.

Key facts to remember

1Matrix addition and subtraction are only possible for matrices of the same order.
2Matrix multiplication AB is defined only if the number of columns of A equals the number of rows of B.
3Matrix multiplication is generally not commutative, i.e., AB ≠ BA.
4Properties of transpose: (A')' = A, (A+B)' = A' + B', (kA)' = kA', (AB)' = B'A'.
5A square matrix A is symmetric if A' = A, and skew-symmetric if A' = -A.
6Any square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix: A = (1/2)(A + A') + (1/2)(A - A').
7A square matrix A is invertible if and only if it is non-singular, i.e., its determinant |A| ≠ 0.
8The inverse of a matrix A is unique, if it exists, and is given by A⁻¹ = (1/|A|) adj(A).

Worked examples

Example 1

Given matrices A = [[2, 3], [1, 0]] and B = [[-1, 2], [0, 5]], find: (i) 2A + B (ii) AB

I**(i) To find 2A + B:**

IIFirst, calculate 2A by multiplying each element of A by 2:

III2A = 2 * [[2, 3], [1, 0]] = [[2*2, 2*3], [2*1, 2*0]] = [[4, 6], [2, 0]]

IVNow, add B to 2A. Since both 2A and B are of order 2x2, addition is possible:

V2A + B = [[4, 6], [2, 0]] + [[-1, 2], [0, 5]] = [[4+(-1), 6+2], [2+0, 0+5]] = [[3, 8], [2, 5]]

VI**(ii) To find AB:**

VIIMatrix A is of order 2x2 and matrix B is of order 2x2. Since the number of columns of A (2) is equal to the number of rows of B (2), matrix multiplication AB is defined and the resulting matrix will be of order 2x2.

VIIIAB = [[2, 3], [1, 0]] * [[-1, 2], [0, 5]]

9Element (1,1) = (2)(-1) + (3)(0) = -2 + 0 = -2

10Element (1,2) = (2)(2) + (3)(5) = 4 + 15 = 19

11Element (2,1) = (1)(-1) + (0)(0) = -1 + 0 = -1

12Element (2,2) = (1)(2) + (0)(5) = 2 + 0 = 2

13Therefore, AB = [[-2, 19], [-1, 2]]

Answer

(i) 2A + B = [[3, 8], [2, 5]] (ii) AB = [[-2, 19], [-1, 2]]

Always check the order of matrices before performing addition/subtraction or multiplication.

Example 2

Express the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] as the sum of a symmetric and a skew-symmetric matrix.

IWe know that any square matrix A can be expressed as the sum of a symmetric and a skew-symmetric matrix, i.e., A = P + Q, where P = (1/2)(A + A') is symmetric and Q = (1/2)(A - A') is skew-symmetric.

IIFirst, find the transpose of A, A':

IIIA' = [[1, 4, 7], [2, 5, 8], [3, 6, 9]]

IVNow, calculate A + A':

VA + A' = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] + [[1, 4, 7], [2, 5, 8], [3, 6, 9]] = [[1+1, 2+4, 3+7], [4+2, 5+5, 6+8], [7+3, 8+6, 9+9]] = [[2, 6, 10], [6, 10, 14], [10, 14, 18]]

VINext, calculate P = (1/2)(A + A'):

VIIP = (1/2) * [[2, 6, 10], [6, 10, 14], [10, 14, 18]] = [[1, 3, 5], [3, 5, 7], [5, 7, 9]]

VIIITo verify P is symmetric, check P' = P:

9P' = [[1, 3, 5], [3, 5, 7], [5, 7, 9]]. Since P' = P, P is a symmetric matrix.

10Now, calculate A - A':

11A - A' = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] - [[1, 4, 7], [2, 5, 8], [3, 6, 9]] = [[1-1, 2-4, 3-7], [4-2, 5-5, 6-8], [7-3, 8-6, 9-9]] = [[0, -2, -4], [2, 0, -2], [4, 2, 0]]

12Next, calculate Q = (1/2)(A - A'):

13Q = (1/2) * [[0, -2, -4], [2, 0, -2], [4, 2, 0]] = [[0, -1, -2], [1, 0, -1], [2, 1, 0]]

14To verify Q is skew-symmetric, check Q' = -Q:

15Q' = [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. Also, -Q = [[0, 1, 2], [-1, 0, 1], [-2, -1, 0]]. Since Q' = -Q, Q is a skew-symmetric matrix.

16Finally, verify A = P + Q:

17P + Q = [[1, 3, 5], [3, 5, 7], [5, 7, 9]] + [[0, -1, -2], [1, 0, -1], [2, 1, 0]] = [[1+0, 3-1, 5-2], [3+1, 5+0, 7-1], [5+2, 7+1, 9+0]] = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

18This is equal to the original matrix A. Hence proved.

Answer

The matrix A can be expressed as P + Q, where P = [[1, 3, 5], [3, 5, 7], [5, 7, 9]] (symmetric) and Q = [[0, -1, -2], [1, 0, -1], [2, 1, 0]] (skew-symmetric).

Remember the formula A = (1/2)(A + A') + (1/2)(A - A') for expressing a matrix as the sum of a symmetric and skew-symmetric matrix.

Example 3

Find the inverse of the matrix A = [[2, 1], [3, 2]] using the adjoint method.

IFirst, calculate the determinant of A, |A|:

II|A| = (2)(2) - (1)(3) = 4 - 3 = 1

IIISince |A| = 1 ≠ 0, the matrix A is non-singular and its inverse exists.

IVNext, find the cofactors of each element:

VC₁₁ = (-1)¹⁺¹ * M₁₁ = 1 * (2) = 2

VIC₁₂ = (-1)¹⁺² * M₁₂ = -1 * (3) = -3

VIIC₂₁ = (-1)²⁺¹ * M₂₁ = -1 * (1) = -1

VIIIC₂₂ = (-1)²⁺² * M₂₂ = 1 * (2) = 2

9Form the cofactor matrix, C:

10C = [[2, -3], [-1, 2]]

11Now, find the adjoint of A, adj(A), which is the transpose of the cofactor matrix:

12adj(A) = C' = [[2, -1], [-3, 2]]

13Finally, use the formula A⁻¹ = (1/|A|) adj(A):

14A⁻¹ = (1/1) * [[2, -1], [-3, 2]] = [[2, -1], [-3, 2]]

Answer

A⁻¹ = [[2, -1], [-3, 2]]

Always check if the determinant is non-zero before attempting to find the inverse. If |A| = 0, the inverse does not exist.

Common mistakes

✗Incorrectly assuming matrix multiplication is commutative (AB = BA).
✗Making errors in calculations during elementary row/column operations, leading to an incorrect inverse.
✗Forgetting to check if the determinant of a matrix is non-zero before attempting to find its inverse; if |A|=0, the inverse does not exist.
✗Confusing the order of matrices in the transpose of a product: (AB)' is B'A', not A'B'.
✗Errors in calculating cofactors or transposing the cofactor matrix to find the adjoint.

Exam tips

★Practice elementary row/column operations extensively to gain speed and accuracy, as they are crucial for finding the inverse.
★Memorise the properties of transpose and inverse, especially (AB)' = B'A' and (AB)⁻¹ = B⁻¹A⁻¹.
★For finding the inverse using the adjoint method, carefully calculate the determinant, cofactors, and adjoint matrix, as a single error can invalidate the entire solution.
★Always show all steps clearly and logically in your solutions, as partial marks are awarded for correct methods even if there's a calculation error.

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