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Matrices – Class 12 Mathematics notes, important questions and MCQs

NCERT Class 12 Mathematics, chapter 3. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A matrix is a rectangular array of numbers. Matrices are added, multiplied and transposed, and are used to solve systems of equations.

Types and operations

Order m × n; square, diagonal, scalar, identity (I), zero matrices.
Addition needs the same order; product AB needs columns of A = rows of B; AB ≠ BA in general.
Transpose: (AB)′ = B′A′; symmetric A′ = A; skew-symmetric A′ = −A (diagonal entries zero).

Useful facts

Every square matrix = symmetric part ½(A + A′) + skew-symmetric part ½(A − A′).
Number of matrices of order m × n with entries from {0, 1} = 2^(mn).
AI = IA = A; inverse exists only for square matrices with non-zero determinant.

Key terms

Symmetric matrix: square matrix equal to its transpose
Skew-symmetric matrix: square matrix with A′ = −A
Elementary operations: row or column interchange, multiplication by a non-zero number, and adding a multiple of one row/column to another

Formulas

(AB)′ = B′A′
A = ½(A + A′) + ½(A − A′)

Important questions with answers

Find x and y if [x + y, 2; 5, x − y] = [6, 2; 5, 2].2 marks

x + y = 6 and x − y = 2, so x = 4, y = 2.

Express A = [3 5; 1 −1] as the sum of a symmetric and a skew-symmetric matrix.3 marks

A′ = [3 1; 5 −1]. P = ½(A + A′) = [3 3; 3 −1] (symmetric). Q = ½(A − A′) = [0 2; −2 0] (skew-symmetric). A = P + Q.

If A = [1 2; 3 4], show that A² − 5A − 2I = 0, and use it to find A⁻¹.5 marks

A² = [1 + 6, 2 + 8; 3 + 12, 6 + 16] = [7 10; 15 22]. 5A = [5 10; 15 20]. A² − 5A = [2 0; 0 2] = 2I, so A² − 5A − 2I = 0. Multiply by A⁻¹: A − 5I − 2A⁻¹ = 0, so A⁻¹ = ½(A − 5I) = ½[−4 2; 3 −1] = [−2 1; 3/2 −1/2]. Check: AA⁻¹ = [1 0; 0 1].

Quick revision

Matrix multiplication is not commutative.
Diagonal of a skew-symmetric matrix is zero.
(A′)′ = A.

Exam tips

Check orders before multiplying: (2 × 3)(3 × 4) gives 2 × 4.
(AB)⁻¹ = B⁻¹A⁻¹ and (AB)′ = B′A′; matrix multiplication is associative but not commutative.

Practice MCQs

Order of a matrix with 2 rows and 3 columns is
1. 3 × 2
2. 6
3. 2 × 3
4. 5
Show answer
Answer: (c) 2 × 3. Rows × columns.
For any square matrix A, AI equals
1. I
2. A′
3. 0
4. A
Show answer
Answer: (d) A. Identity.
If A is 2 × 3 and B is 3 × 4, then AB is
1. 2 × 4
2. 3 × 3
3. 4 × 2
4. not defined
Show answer
Answer: (a) 2 × 4. Outer orders.
Diagonal elements of a skew-symmetric matrix are
1. 0
2. 1
3. any number
4. equal
Show answer
Answer: (a) 0. aᵢᵢ = −aᵢᵢ.
If A = [2 0; 0 3], then A² is
1. [2 0; 0 3]
2. [4 0; 0 6]
3. [4 0; 0 9]
4. [0 4; 9 0]
Show answer
Answer: (c) [4 0; 0 9]. Square the diagonal.
Number of 2 × 2 matrices with entries 0 or 1 is
1. 4
2. 8
3. 16
4. 32
Show answer
Answer: (c) 16. 2⁴.
(AB)′ equals
1. A′B′
2. AB
3. B′A′
4. BA
Show answer
Answer: (c) B′A′. Reversal law.
If A is a 3 × 3 matrix with A² = A, then (I − A)² equals
1. I
2. A
3. 0
4. I − A
Show answer
Answer: (d) I − A. I − 2A + A² = I − A.

All chapters of Class 12 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.