Show that the inverse of a square matrix A exists if and only if the
eigenvalues λ1
,λ2
,··· ,λn of A are different from zero. If A
−1
exists
show that its eigenvalues are 1
λ1
,
1
λ2
,···
1
λn
.
If is an eigenvalue of the matrix A, and is a corresponding column-eigenvector, then the inverse matrix doesn't exist, since .
Conversely, if is not an eigenvalue of the matrix A, then is not a root of the characteristic polynomial of the matrix A. Therefore, and the matrix A is invertible.
Assuming A is invertible, consider the characteristic polynomial of the matrix A.
From this we can see that is an eigenvalue of the matrix A if and only if is an eigenvalue of the matrix . Moreover, one can see that multiplicity of the eigenvalue of the matrix A is the same as multiplicity of the eigenvalue of the matrix .
So, if are eigenvalues of the matrix A, then are eigenvalues of the matrix .
The assertion is proved.
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