Question

Suppose P is invertible and PBP^-1 . Solve for B in terms of A

Answer 1

I'm supposing you meant A=PBP^-1.

Matrix multiplication is NOT commutative. But you can multiply both sides of an equation by a matrix as long as you keep the matrix on the same side on both sides. Wait what? How about an illustration:

Let A, B, and X be matrices with A=B.
Then AX = BX and XA = XB, but you can't say AX = XB or XA = BX.

Having said that, PP^-1 = P^-1P = I, and AI = IA = A, so there it kinda looks commutative.

So multiply both sides of the equation by P and P^-1 in the right places, and you'll solve for B.