Let the eigenvalues of matrix A of order 3 be the eigenvectors corresponding to 1,2, - 3, A1, A2 and A3 in turn. Let the square matrix B = a * - 2A + 3I, find the eigenvalues of B ^ - 1 and det (b ^ - 1) Please answer for me,

Let the eigenvalues of matrix A of order 3 be the eigenvectors corresponding to 1,2, - 3, A1, A2 and A3 in turn. Let the square matrix B = a * - 2A + 3I, find the eigenvalues of B ^ - 1 and det (b ^ - 1) Please answer for me,

Because the eigenvalues of a matrix of order 3 are 1,2, - 3
So | a | = 1 * 2 * (- 3) = - 6
If λ is the eigenvalue of a and a is the eigenvector of a, then AA = λ a
Multiply a * left on both sides to get λ a * a = a * AA = | a | a
So when λ ≠ 0, a * a = (| a | / λ) a
So Ba = a * a - 2AA + 3A = (|a | / λ - 2 λ + 3) a
So the eigenvalue of B is: | a | / λ - 2, λ + 3
The eigenvalues of a are 1,2, - 3, | a | = - 6
The eigenvalues of B are - 5, - 4,11
So | B | = (- 5) * (- 4) * 11 = 220