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# Invertible matrix

Consider the matrices and . If is invertible prove that is also invertible.

Solution

So we have to answer the question if is a zero of the essentially same characteristic polynomials. and have quite similar characteristic polynomials. In fact if denotes the polynomial of , then the polynomial of will be . It is easy to see that cannot be an eigenvalue of the matrix, otherwise it wouldn’t be invertible. Now, let us assume that is not invertible. Then it must have an eigenvalue of and let be the corresponding eigenvector. Hence:

meaning that has an eigenvalue of which is a contradiction. The result follows.