If A and B are invertible matrices, then which of the following is not correct?

Question

Philoid · Accepted Answer

Given A and B are invertible matrices.

Consider (AB) B^-1 A^-1

⇒ (AB) B^-1 A^-1 = A(BB^-1) A^-1

= AIA^-1 = (AI) A^-1

= AA^-1 = I

⇒ (AB)^-1 = B^-1 A^-1 … option (C)

Also AA^-1 = I

⇒ |AA^-1| = |I|

⇒ |A| |A^-1| = 1

∴ det (A)^-1 = [det (A)]^-1 … (B)

We know that

⇒ adj A = |A|. A^-1 … option (A)

But

∴ (A + B)^-1 ≠ B^-1 + A^-1

Hence, option (D) is not correct.