Question

Using any example of a 2 by 2 matrix;

Show that (A inverse) inverse = A; where A is a 2 by 2 matrix

Answer 1

if A is a 2x2 matrix, it looks like this (see picture where A = …). Note that (a and b) and (b and c) are diagonal from one another.
The inverse of A is A^-1 (see picture), where A^-1 = (1/(determinant of A))*(adjoint of A). Note that the inverse of A only exists if ad - bc ≠ 0.

So, if ANY example of a 2 by 2 matrix is allowed… we use an example where a = 1, b = 2, c = -3, d = -5. The inverse matrix will have the terms -5, -2, 3, 1. (See photo)

To show that the inverse OF the inverse equals A, follow the same process. And.. yes, (A^-1)^-1 = A. We want to see the identity matrix 1, 0, 0, 1 since A*A^-1 = I (this is an uppercase i).

I’m on my phone so it’s harder to type. Refer to the picture to see my work:

I hope this helps!