Question

Let M⁴ = I, (where I denotes the identity matrix) and M ≠ I, M² ≠ I and M³ ≠ I. Then, for any natural number k, M⁻¹ equals:
1. M^{4k + 1}
2. M^{4k + 2}
3. M^{4k + 3}
4. M^4k

Answer 1

Correct Answer - Option 3 : M^{4k + 3}

Explanation:

Let M⁴ = I      ---(i)

Multiplying by M^-1 on both sides

M⁴ M^-1 = I – M^-1

M³ = M^-1      ---(ii)

Multiplying M⁴ on both sides in equation (i)

M⁴⋅ M⁴ = I ⋅ M⁴

M⁸ = M⁴      ---(iii)

Multiplying M^-1 on both sides, we get:

M⁸ – M^-1 = M⁴ M^-1

M⁷ = M³      ---(iv)

From (ii) and (iv)

M^-1 = M³ = M⁷

Now again multiplying M⁴ on both sides in equation (iii)

M⁸⋅ M⁴ = M⁴ M⁴

M¹² = M⁸

M¹² = M⁴ [from (iii)]

M¹² = I      ---(v) [from (i)]

Multiplying M^-1 on both sides we get

M¹¹ = M^-1      ---(vi)

From (ii), (iv) and (vi)

M^-1 = M³ = M⁷ = M¹¹ = …. & So on

3, 7, 11 ….. from general term 4k + 3 where k = 0, 1, 2

Hence, M^-1 = M^{4k + 3}

Option C is correct.