- #1

Mr Davis 97

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## Homework Statement

Let ##A## and ##B## be ##n \times n## matrices

1) Suppose ##A^2 = 0##. Prove that ##A## is not invertible.

2) Suppose ##AB=0##. Could ##A## be invertible.

3) If ##AB## is invertible, then ##A## and ##B## are invertible

## Homework Equations

## The Attempt at a Solution

1) Suppose that ##A## were invertible. Then ##A^2 = 0 \Rightarrow A = 0##. This is a contradiction, because we know that the zero matrix is not invertible.

So I think this is one way of doing it. What are some others?

2) It seems that we can proceed in a similar way as 1). Assume that ##A## is invertible, then ##B=0##. Thus, since ##A## is can be an arbitrary matrix, it can be invertible.

Is this the correct way of doing this?

3) This is the one that I am having the most trouble with. Assume that ##AB## is invertible. So we know that there exists a ##n \times n## matrix ##C## such that ##C(AB) = (AB)C = I_n##. So ##(CA)B = A(BC) = I_n##. So we can conclude that ##B## has a left-inverse and that ##A## has a right-inverse. But I don't see how we can conclude that ##B## has a right-inverse and ##A## has a left-inverse.