Class 12 Maths Chapter 3: Matrices

Matrices is Chapter 3 of NCERT Class 12 Maths. It introduces matrices as rectangular arrangements of numbers, the different types of matrices, and the operations of addition, scalar multiplication, multiplication and transpose, along with symmetric, skew-symmetric and invertible matrices. These notes, formulas and solved examples will help you score full marks on this short but scoring chapter in the CBSE board exam.

What you will learn

The order of a matrix and how to construct a matrix from a rule for aᵢⱼ
Types of matrices such as row, column, square, diagonal, scalar, identity and zero matrices
Addition, scalar multiplication and multiplication of matrices with their properties
The transpose of a matrix and symmetric and skew-symmetric matrices
What an invertible matrix is and why its inverse is unique

Key Concepts and Formulas

Matrix Basics and Types

A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has order m × n and mn elements.

Order m × n: m rows, n columns
aᵢⱼ = element in row i, column j
Identity matrix of order 2:
I = [ 1  0 ]
    [ 0  1 ]
A = B ⇔ same order and aᵢⱼ = bᵢⱼ for all i, j

Square matrix: number of rows = number of columns.
Diagonal matrix: a square matrix whose non-diagonal elements are all zero.
Scalar matrix: a diagonal matrix with all diagonal elements equal.
Identity matrix I: a diagonal matrix with every diagonal element 1.
Zero matrix O: every element is 0.

Operations on Matrices

Two matrices can be added or subtracted only if they have the same order. Multiplication follows a row-into-column rule: each element of AB is found by multiplying a row of A with a column of B and adding.

(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ   (same order only)
(kA)ᵢⱼ = k × aᵢⱼ
A (m × n) × B (n × p) = AB (m × p)
(AB)ᵢₖ = Σ aᵢⱼ bⱼₖ
A + B = B + A
(AB)C = A(BC)
A(B + C) = AB + AC
AI = IA = A
In general AB ≠ BA

Row by column multiplication of a 2 by 3 matrix and a 3 by 2 matrix, Class 12 Maths Matrices — Each entry of AB is a row times a column

AB is defined only if the number of columns of A equals the number of rows of B.
AB = O does not mean A = O or B = O.

Transpose of a Matrix

The transpose Aᵀ (also written A′) is obtained by changing rows into columns.

A is m × n ⇒ Aᵀ is n × m
(Aᵀ)ᵀ = A
(A + B)ᵀ = Aᵀ + Bᵀ
(kA)ᵀ = kAᵀ
(AB)ᵀ = BᵀAᵀ

Transpose of a 2 by 3 matrix where rows become columns, Class 12 Maths Chapter 3 Matrices — Rows of A become the columns of Aᵀ

In (AB)ᵀ the order is reversed.

Symmetric and Skew-Symmetric Matrices

These are special square matrices defined using the transpose. Every square matrix can be split into a symmetric part and a skew-symmetric part.

Symmetric:      Aᵀ = A,  aᵢⱼ = aⱼᵢ
Skew-symmetric: Aᵀ = −A, aᵢⱼ = −aⱼᵢ
                so every aᵢᵢ = 0
A = ½(A + Aᵀ) + ½(A − Aᵀ)
    symmetric   skew-symmetric

All diagonal elements of a skew-symmetric matrix are 0.

Invertible Matrices

A square matrix A is invertible if there is a square matrix B of the same order with AB = BA = I. Then B is called the inverse of A and is written A⁻¹.

AB = BA = I ⇒ B = A⁻¹
(AB)⁻¹ = B⁻¹A⁻¹

The inverse of a square matrix, if it exists, is unique.

Solved Examples

Example 1: Construct a 2 × 3 matrix A = [aᵢⱼ] with aᵢⱼ = 2i − j.

a₁₁ = 2 − 1 = 1    a₁₂ = 2 − 2 = 0
a₁₃ = 2 − 3 = −1   a₂₁ = 4 − 1 = 3
a₂₂ = 4 − 2 = 2    a₂₃ = 4 − 3 = 1
∴ A = [ 1  0  −1 ]
      [ 3  2   1 ]

Example 2: Find x and y from the matrix equation in the working below.

[ x + y    4   ]   [ 9  4 ]
[   6    x − y ] = [ 6  3 ]
x + y = 9 and x − y = 3
Adding: 2x = 12 ⇒ x = 6
y = 9 − 6 = 3
∴ x = 6, y = 3

Example 3: For the matrices A and B given in the working, find 2A − 3B.

A = [ 3  −1 ]    B = [ 1   2 ]
    [ 2   5 ]        [ 0  −1 ]
2A = [ 6  −2 ]   3B = [ 3   6 ]
     [ 4  10 ]        [ 0  −3 ]
∴ 2A − 3B = [ 3  −8 ]
            [ 4  13 ]

Example 4: For the matrices A and B given in the working, find AB and BA and check whether they are equal.

A = [ 2  1 ]    B = [ 1  −1 ]
    [ 0  3 ]        [ 4   2 ]
AB = [ 2 + 4    −2 + 2 ] = [  6  0 ]
     [ 0 + 12    0 + 6 ]   [ 12  6 ]
BA = [ 2 + 0   1 − 3 ] = [ 2  −2 ]
     [ 8 + 0   4 + 6 ]   [ 8  10 ]
∴ AB ≠ BA

Example 5: For the matrix A in the working, show that A² − 2A − 3I = O.

A = [ 1  2 ]
    [ 2  1 ]
A² = [ 1 + 4   2 + 2 ] = [ 5  4 ]
     [ 2 + 2   4 + 1 ]   [ 4  5 ]
2A = [ 2  4 ]    3I = [ 3  0 ]
     [ 4  2 ]         [ 0  3 ]
A² − 2A − 3I = [ 5 − 2 − 3   4 − 4 − 0 ]
               [ 4 − 4 − 0   5 − 2 − 3 ]
∴ A² − 2A − 3I = O

Example 6: Express the matrix A in the working as the sum of a symmetric and a skew-symmetric matrix.

A = [ 4   6 ]    Aᵀ = [ 4   2 ]
    [ 2  −3 ]         [ 6  −3 ]
P = ½(A + Aᵀ) = ½ [ 8   8 ] = [ 4   4 ]
                  [ 8  −6 ]   [ 4  −3 ]
Q = ½(A − Aᵀ) = ½ [  0  4 ] = [  0  2 ]
                  [ −4  0 ]   [ −2  0 ]
Pᵀ = P (symmetric), Qᵀ = −Q (skew-symmetric)
∴ A = P + Q

Example 7: Show that the matrix B in the working is the inverse of A.

A = [ 3  2 ]    B = [  3  −2 ]
    [ 4  3 ]        [ −4   3 ]
AB = [ 9 − 8     −6 + 6 ] = [ 1  0 ]
     [ 12 − 12   −8 + 9 ]   [ 0  1 ]
Similarly BA = I
∴ A⁻¹ = B

Common Mistakes to Avoid

Always check the orders before multiplying; if columns of A ≠ rows of B, write "AB is not defined".
Do not assume (A + B)² = A² + 2AB + B²; for matrices it equals A² + AB + BA + B² because AB may not equal BA.
In (AB)ᵀ remember to reverse the order to BᵀAᵀ; writing AᵀBᵀ is a very common error.
When adding a matrix and a number, such as A − 3, write it as A − 3I; a scalar alone cannot be added to a matrix.

Quick Revision

A (m × n) × B (n × p) = AB (m × p)
In general AB ≠ BA
(AB)ᵀ = BᵀAᵀ
(AB)⁻¹ = B⁻¹A⁻¹
Symmetric: Aᵀ = A
Skew-symmetric: Aᵀ = −A, diagonal = 0
A = ½(A + Aᵀ) + ½(A − Aᵀ)