Notes from Section 8

Prove: A is not invertible if and only if 0 is an eigenvalue of A. Since this is an if and only if statement, we need to prove bothimplications, that means you have to prove:

1) If A is not invertible then 0 is an eigenvalue.2) If 0 is an eigenvalue, then A is not invertible.Proof of 1) Assume A is not invertible. Then Ax = 0 does not have only trivial solution by invertible matrix theorem.

Since it does have the trivial solution (letting x = 0 gives a solution), but not only the trivial solution, there must be some othersolution. Thus the equation Ax = 0 = 0x has a nontrivial solution, so by definition, 0 is an eigenvalue.

Proof of 2) Assume 0 is an eigenvalue. Thus there is some nontrivial solution to Ax = 0x = 0. By the invertible matrixtheorem, if A was invertible there would only be the trivial solution. Since there in a nontrivial solution, it must be the casethat A is not invertible.

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