The easiest way to solve this problem is using the spectral theorem!!
The transpose matrix is just when the rows from left to right turn into the columns up to down from top to bottom of given matrix A.
Since we are dealing with a n by n matrix A then the transpose matrix A is also n by n !!
And it looks like from the given info that this is a map to itself since the transpose matrix A is just the same as the matrix A in dimension and value with the rows turning into the columns left to right up to down like previously mentioned!
This map is going to be reflecting about y=x line so they should have eigenvalues 1 and -1 respectively!
Since the transpose matrix A is also equal to the inverse matrix A (because it is an orthogonal matrix from the given info)