Findmatrix P such that P'AP orthogonally diagonalizes A. Verify that P'AP gives the proper diagonal form. (Enter each matrix in the form [[row 1], [row 2], ...], where each row is a comma-separated list.)5 200250 00 05 2A =0 0 25(woo) (o020) ovwo)(후마무미(부-부 미r우or 조시or슨or푸] )(P, PTAP) =|

The given matrix is A = 5200250000520025. Note that A is a block matrix of the form A = A100A2, where Ai=5225.By property of block matrices, we know that eigenvalues of A can be estimated from the eigenvalues of the matrices of the blocks. Consider the matrix Ai=5225. Let λ be the eigenvalue of Ai. Then, we have,Ai-λI=0⇒5-λ225-λ=0⇒(5-λ)2-4=0⇒25+λ2-10λ-4 = 0⇒λ2-10λ+21=0⇒λ(λ-3)-7(λ-3)=0⇒(λ-3)(λ-7)=0⇒λ=3, 7Thus, the eigenvalues of A are 3, 3, 7, 7.Next, we find the eigenvectors, corresponding to each eigenvalue.For λ=7,Let v = v1v2v3v4-22002-20000-22002-2v1v2v3v4=0000-1100000000-110000v1v2v3v4=0000⇒-v1+v2=0; -v3+v4=0.⇒v1=v2; v3=v4Let v1=v2=t; v3=v4=s.Therefore, v = ttss=t1100+s0011.For λ=3,Let v = v1v2v3v42200220000220022v1v2v3v4=00001100000000110000v1v2v3v4=0000⇒v1+v2=0; v3+v4=0.⇒v1=-v2; v3=-v4Let v1=t; v3=s.Therefore, v = t-ts-s=t1-100+s001-1. Now, Define the matrix P as P = 101010-100101010-1and matrix D as D=7000070000300003.Therefore, we have diagonalized the matrix A as there exists matrix P and D such thatP-1AP = D. Next, we have to verify that PTAP is a diagonal matrix.PTAP = 110000111-100001-15200250000520025101010-100101010-1 =770000773-300003-3101010-100101010-1 =140000140000600006Hence, we can see that PTAP is a diagonal matrix