Suppose A and B are n times n matrices with B and are invert

Suppose A and B are n times n matrices with B and are invertible. Show A is invertible.

Solution

We know that Multiplication of matrices is associative.

I = (AB)(AB)^-1

= A(B(AB)^-1)

Multiply both sides by A^-1

we get A^-1 = B(AB)^-1

Now AB is invertible and AB^-1 exists , B is not equal to zero

So, A^-1 exists

Henec A is invertible