Eigenvalue Assignment, 3 × 3 3 \times 3 Matrix

Algebra Level 3

Let

A = [ 1 5 0 2 3 1 1 1 2 ] A = \begin{bmatrix} 1 && 5 && 0\\ -2 && 3 && 1 \\ 1 && 1 && 2 \end{bmatrix}

and let

B = A [ 2 3 1 ] [ k 1 k 2 k 3 ] B = A - \begin{bmatrix} 2 \\ 3 \\1 \end{bmatrix} \begin{bmatrix} k_1 && k_2 && k_3 \end{bmatrix}

Find k 1 , k 2 , k 3 k_1 , k_2 , k_3 that will place the eigenvalues of matrix B B at 5 , 3 , 2 -5 , -3, -2 , then enter 3 ( k 1 + k 2 + k 3 ) 3 (k_1 + k_2 + k_3) as your answer.


The answer is 32.

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