Question

If A is a non-singular square matrix such that A³ = I, then A^-1 = _______________.

Answer 1

Given: A³ = I


A³ = I
Multiplying both sides by A⁻¹, we get
⇒ A³A⁻¹ = I A⁻¹
⇒ A²(AA⁻¹) = I A⁻¹
⇒ A²(I) = A⁻¹
⇒ A² = A⁻¹

Hence, if A is a non-singular square matrix such that A³ = I, then A⁻¹ = A².