Let σ(A) be the set of all eigenvalues of A. Show that σ(A)=σ(AT) where AT is the transpose matrix of A.

**Answer**

The matrix (A−λI)T is the same as the matrix (AT−λI), since the identity matrix is symmetric.

Thus:

det(AT−λI)=det((A−λI)T)=det(A−λI)

From this it is obvious that the eigenvalues are the same for both A and AT.

**Attribution***Source : Link , Question Author : Zizo , Answer Author : Invisible*