Question

Prove that (A^–1)′ = (A′)^–1, where A is an invertible matrix.

Answer 1

Since A is an invertible matrix, so it is non-singular

We know that |A| = |A′|.

But |A| ≠ 0.

So |A′| ≠ 0 i.e. A′ is invertible matrix.

Now we know that AA^–1 = A^–1A = I.

Taking transpose on both sides, we get

(A^–1)′A′ = A′(A^–1)′

= (I)′

= I

Hence (A^–1)′ is inverse of A′

i.e., (A′)^–1 = (A^–1)