Diagonalize Matrix A

Question

I am told to diagonalize Matrix A i solved for P and P inverse

     A= 2  4 1        P= 1  4/5  5               P^-1 =   1   -6  -4/5
        0 -3 5           0   0   1                        0   5/4  1
        0  0 1           0   1  -5/4                      0    1   0 

Now am i suppose to do the following P-1AP = D where D is the diagonal Matrix do I then apply D.A to get the the diagonalized matrix? or do i do the following A = PDP-1?

Answer 1

You have diagonalized the matrix. That is, the expression $PDP^{-1} = A$ is the "diagonalization" of the matrix $A$.

Answer 2

just find the eigen values of $A$ which in your case is $2,-3,1$ and the corresponding eigen vectors the columns of $P$ will be the three eigen vectors

Answer 3

You're given all the information you need. If you look at your matrix A, you'll notice that it is in upper triangular form. Recall, that if your matrix A is in upper triangular form, then the eigenvalues of A are simply the diagonal entries. Thankfully, the eigenvectors corresponding to each $\lambda$ value have already been calculated and given to you by your matrix P. Furthermore, a nice and easy way to check if $P^{-1}$AP = D is to check if AP = DP.