LIWA

Matrices

ICSE Class 10 · Mathematics · Chapter 9

What this chapter covers

Key points

Questions answered

Define the order of a matrix. What is the order of a matrix with 3 rows and 2 columns?
The order of a matrix is the number of its rows followed by the number of its columns, written m × n. A matrix with 3 rows and 2 columns has order 3 × 2.
A matrix has 6 elements. Write down all its possible orders.
1 × 6, 6 × 1, 2 × 3 and 3 × 2.
State the difference between a null matrix and an identity matrix of order 2.
Every element of a null matrix is 0, giving beginbmatrix 0 & 0 \ 0 & 0 endbmatrix. The identity matrix has 1 on the leading diagonal and 0 elsewhere, giving beginbmatrix 1 & 0 \ 0 & 1 endbmatrix. The null matrix acts like 0 in addition; the identity matrix acts like 1 in multiplication.
A is of order 2 × 3 and B is of order 3 × 2. Does A + B exist? Do AB and BA exist? Give the orders where they do.
A + B does not exist, since the orders differ. AB exists with order 2 × 2, and BA exists with order 3 × 3.
Give a reason why a column matrix of order 2 × 1 cannot be multiplied on the right by a 2 × 2 matrix, that is, why CA does not exist when C is 2 × 1 and A is 2 × 2.
For CA to exist, the number of columns of C must equal the number of rows of A. C has 1 column while A has 2 rows, so the matrices are not compatible for this product.
If A = beginbmatrix 4 & -2 \ 1 & 3 endbmatrix and B = beginbmatrix -1 & 5 \ 2 & 0 endbmatrix, find 2A + B.
2A = beginbmatrix 8 & -4 \ 2 & 6 endbmatrix, so 2A + B = beginbmatrix 7 & 1 \ 4 & 6 endbmatrix.
Find the matrix X if 2X = beginbmatrix 1 & 3 \ 5 & -1 endbmatrix + beginbmatrix 3 & 1 \ -1 & 5 endbmatrix.
The sum is beginbmatrix 4 & 4 \ 4 & 4 endbmatrix, so X = (1)/(2)beginbmatrix 4 & 4 \ 4 & 4 endbmatrix = beginbmatrix 2 & 2 \ 2 & 2 endbmatrix.
Write beginbmatrix 3 & -6 \ 9 & 0 endbmatrix as 3 times another matrix, and find (1)/(3) of the matrix.
beginbmatrix 3 & -6 \ 9 & 0 endbmatrix = 3beginbmatrix 1 & -2 \ 3 & 0 endbmatrix, so one-third of the matrix is beginbmatrix 1 & -2 \ 3 & 0 endbmatrix.

Past board exam questions

ICSE 2027 sample paper · 1 mark

Given two matrices A and B, where A = beginbmatrix -1 & 2 endbmatrix and B = beginbmatrix 3 \ 4 endbmatrix then product AB is:

  1. beginbmatrix -3 \ 8 endbmatrix
  2. 5
  3. beginbmatrix 5 endbmatrix
  4. beginbmatrix -3 & 8 endbmatrix
Show answer
beginbmatrix 5 endbmatrix

ICSE 2027 sample paper · 3 marks

Given, matrix A = beginbmatrix x & 1 \ 2 & 2 endbmatrix and B = beginbmatrix x \ x - 2 endbmatrix, such that AB is a null matrix. Find the: (a) order of the null matrix. (b) value of x, given that x > 0.

Show answer
(a) A is 2 × 2 and B is 2 × 1, so AB (the null matrix) is of order 2 × 1. (b) AB = beginbmatrix x · x + 1(x - 2) \ 2x + 2(x - 2) endbmatrix = beginbmatrix x2 + x - 2 \ 4x - 4 endbmatrix = beginbmatrix 0 \ 0 endbmatrix 4x - 4 = 0 ⇒ x = 1; check: 1 + 1 - 2 = 0 ✓ (and x2 + x - 2 = 0 gives x = 1 or -2; x > 0). So x = 1.

ICSE 2026 sample paper · 1 mark

If matrix A = beginbmatrix -1 & 2 endbmatrix and matrix B = beginbmatrix 3 \ 4 endbmatrix, then matrix AB is equal to:

  1. beginbmatrix -3 endbmatrix
  2. beginbmatrix 8 endbmatrix
  3. beginbmatrix 5 endbmatrix
  4. beginbmatrix -1 & 2 \ 3 & 4 endbmatrix
Show answer
beginbmatrix 5 endbmatrix

ICSE 2026 board paper · 1 mark

Given matrix A = beginbmatrix 2 & 3 \ 1 & 2 endbmatrix and matrix B = beginbmatrix 2 & -4 endbmatrix. Product AB is a matrix of order:

  1. 2 × 2
  2. 2 × 1
  3. 1 × 2
  4. product AB is not possible
Show answer
product AB is not possible

ICSE 2026 board paper · 4 marks

Given, A = beginbmatrix 3 & 1 \ 5 & 3 endbmatrix and B = beginbmatrix -1 & a \ 3 & -5 endbmatrix and product AB = beginbmatrix b & 7 \ 4 & 5 endbmatrix. Find the values of 'a' and 'b'.

Show answer
AB = beginbmatrix 3(-1) + 1(3) & 3a + 1(-5) \ 5(-1) + 3(3) & 5a + 3(-5) endbmatrix = beginbmatrix 0 & 3a - 5 \ 4 & 5a - 15 endbmatrix Comparing with beginbmatrix b & 7 \ 4 & 5 endbmatrix: b = 0; 3a - 5 = 7 ⇒ a = 4 (check: 5(4) - 15 = 5 ✓). a = 4, b = 0.

ICSE 2025 sample paper · 1 mark

If A = beginbmatrix 3 & -2 endbmatrix and B = beginbmatrix -1 & 4 \ 2 & 0 endbmatrix Assertion (A): Product AB of the two matrices A and B is possible. Reason (R): Number of columns of matrix A is equal to number of rows in matrix B.

  1. A is true, R is false.
  2. A is false, R is true.
  3. Both A and R are true, and R is the correct reason for A.
  4. Both A and R are true, and R is incorrect reason for A.
Show answer
Both A and R are true, and R is the correct reason for A.

ICSE 2025 board paper · 1 mark

If A = beginbmatrix 0 & 1 \ 1 & 0 endbmatrix, then A2 is equal to:

  1. beginbmatrix 1 & 1 \ 0 & 0 endbmatrix
  2. beginbmatrix 0 & 0 \ 1 & 1 endbmatrix
  3. beginbmatrix 1 & 0 \ 0 & 1 endbmatrix
  4. beginbmatrix 0 & 1 \ 1 & 0 endbmatrix
Show answer
beginbmatrix 1 & 0 \ 0 & 1 endbmatrix

ICSE 2025 compartment paper · 1 mark

If matrix A of order 2 × 1 and matrix B of order 2 × 2 are added, then the order of the matrix A + B is:

  1. 2 × 2
  2. 2 × 1
  3. 1 × 2
  4. A + B is not possible
Show answer
A + B is not possible