Invertibility & Isomorphism

**leungsta** · October 26th, 2008, 05:54 PM

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.

**ThePerfectHacker** · October 26th, 2008, 08:08 PM

Originally Posted by leungsta

a) Suppose that A^2 = O. Prove that A is not invertible

If $AA^{-1} = I \implies A^2A^{-1} = A \implies 0 = A$ a contradiction.

b) Suppose that AB = O for some nonzero n x n matrix B. Could A be invertible? Explain.

If $A$ is invertible then $A^{-1} (AB) = A^{-1} (0) \implies B = 0$ a contradiction.

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