Let A be a real n imes n constant matrix, let
lambda _(1),lambda _(2),lambda _(3),cdotscdots,lambda _(n)
represent all the eigenvalues of the matrix A. Consider the matrix
B=^( def )I+2A+A^(2)-(lambda _(1)+1)^(2)I
Which of the following statements about the matrix B is true?
(A) Tr(B)=lambda _(1)^(2)+lambda _(2)^(2)+lambda _(3)^(2)+cdots+lambda _(n)^(2)
(B) det(B)=(lambda _(1)lambda _(2)lambda _(3)cdotslambda _(n))^(2)
(C) The eigenvectors of the matrix B
are exactly the same as the eigenvectors of A
(D) The eigenvalues of the matrix B are
(lambda _(1)-1)^(2),(lambda _(2)-1)^(2),(lambda _(3)-1)^(2),cdots,(lambda _(n)-1)^(2)
30. Let A be a real n n constant matrix, let
X1,
2
represent all the eigenvalues of the matrix A. Consider the matrix
B def I+2A+A2 - (A1+1)2I
Which of the following statements about the matrix B is true?
(A) Tr (B)=+X+X3+...+x2 (B) det(B)=(A1A2A3...Xn)2 (C) The eigenvectors of the matrix B are exactly the same as the eigenvectors of A (D) The eigenvalues of the matrix B are (1 - 1)2, J2 - 1)2, (3-1)2
An -1)2