12. For each matrix A with real valued entries, find a diagonal matrix D and an invertible matrix Q suchthat QT AQ = D.(a) ()(b) ()31 214020 -1

Part(a): Given matrix is, A=1332 Now, finding the eigenvalues of the given matrix, A-λI=01-λ332-λ=01-λ2-λ-9=02-2λ-λ+λ2-9=0λ2-3λ-7=0λ=3±372 So, the eigenvalues are λ1=3+372 and λ2=3-372 So, the diagonal matrix is, D=3+372003-372Now, the corresponding eigenvectors from the obtained values are, For λ1=3+372 the corresponding eigenvector is -1+37 6 And λ2=3-372 the corresponding eigenvector is -1-37 6 So, the invertible matrix Q is, Q=-1+37-1-3766 Now, QT of the matrix Q is, QT=Q-1QT=1237--1-371237-1237-1+371237 Therefore, QTAQ=1237--1-371237-1237-1+3712371332-1+37-1-3766=3+372003-372QTAQ=D Part(b): Given matrix is, A=31214020-1 Now, finding the eigenvalues of the given matrix, A-λI=03-λ1214-λ020-1-λ=03-λ4-λ-1-λ-1-1-λ-0+224-λ=0λ=3,3+352,3-352 So, the eigenvalues are, λ1=3,λ2=3+352 and λ33-352 So, the diagonal matrix is, D=30003+3520003-352Now, the corresponding eigenvectors from the obtained values are, λ1=3 is 2-21λ2=3+352 is 5+357+354 And λ3=3-352 is -35+57-354 So, the invertible matrix Q is, Q=25+35-35-27+357-35144 Now, QT of the matrix Q is, QT=Q-1Q=29-29195-1365+11805-2545-1+5365-13655+295 Therefore, QAQT=29-29195-1365+11805-2545-1+5365-13655+29531214020-125+35-35-27+357-35144QAQT=D