Question

The number of matrices A = `[(a, b),(c, d)]`, where a, b, c, d ∈ {–1, 0, 1, 2, 3, ............, 10} such that A = A^–1, is ______.

Options

• 20

• 30

• 40

• 50

Answer 1

The number of matrices A = `[(a, b),(c, d)]`, where a, b, c, d ∈ {–1, 0, 1, 2, 3, ............, 10} such that A = A^–1, is 50.

Explanation:

Given matrix is A = `[(a, b),(c, d)]` and A = A^–1

Hence, A² = A . A^–1 = I

`\implies [(a, b),(c, d)][(a, b),(c, d)] = [(1, 0),(0, 1)]`

`\implies [(a^2 + bc, ab + bd),(ac + cd, bc + d^2)] = [(1, 0),(0, 1)]`

Compare the corresponding elements of the above matrix.

∴ a² + bc = 1  ...(i)

ab + bd = 0  ...(ii)

ac + cd = 0  ...(iii)

bc + d² = 1  ...(iv)

From (i) and (iv),

a² – d² = 0

`\implies` (a + d) = 0 or a – d = 0

Case – I:

a + d = 0

`\implies` (a, d) = (–1, 1), (0, 0), (1, –1)

(a) (a, d) = (–1, 1) From (i),

1 + bc = 1 `\implies` bc = 0

b = 0, c = 12 possibilities

c = 0, b = 12 possibilities

 Here, (0, 0) is repeated the total possibilities are 2 × 12 = 24

Total pairs = 24 – 1 = 23.

(b) (a, d) = (–1, 1) `\implies` bc = 0 `rightarrow` 23 pairs

(c) (a, d) = (0, 0) `\implies` bc = 1

`\implies` (b, c) = (1, 1) and (–1, 1), 2 pairs

Case – II:

Here, a = d

From (ii) and (iii),

if a ≠ 0 then b = c = 0

a² = 1

a = ±1 = d

(a, d) = (1, 1), (–1, –1) `rightarrow` 2 pairs

Total number of pairs = 23 + 23 + 2 + 2 = 50 pairs