(1) if A and B are nonsingular matrics, which, if any of the following must be nonsingular?

(a) A+B (b) (AB)^2 (c) kA (k subeset R)

justify your answer either be obtaining the inverse or by a counter example to show it need not exist.

(2) if A,B and C are square matrices, such that AB = In and CA = In, prove that A is nonsigular and B = C = A^-1.

(Hint: try multiplying AB and CA by C and B respectively)

For (1), I know (b) and (c) is true,

(c) is true iff
how about (a)?

What about A = I and B = -I?
And I dunno how to start (2)