Let A be an n x n matrix.
a) Suppose that A^2 = O. Prove that A is not invertible
b) Suppose that AB = O for some nonzero n x n matrix B. Could A be invertible? Explain.
If $\displaystyle AA^{-1} = I \implies A^2A^{-1} = A \implies 0 = A$ a contradiction.
If $\displaystyle A$ is invertible then $\displaystyle A^{-1} (AB) = A^{-1} (0) \implies B = 0$ a contradiction.b) Suppose that AB = O for some nonzero n x n matrix B. Could A be invertible? Explain.