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Let A be a 3*3 matrix whose characteristics roots are 3,2,-1. If $B=A^2-A$ then |B|=?

a)24

b)-2

c)12

d)-12

Please explain in detail.
in Linear Algebra by (259 points) | 157 views
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24

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Best answer
Let

A = $\begin{bmatrix} 3 &0 &0 \\ 0&2 &0 \\ 0&0 &-1 \end{bmatrix}$

A$^{2}$ = $\begin{bmatrix} 9 &0 &0 \\ 0&4 &0 \\ 0&0 &1 \end{bmatrix}$

B = A$^{2}$ - A

    =$\begin{bmatrix} 6 &0 &0 \\ 0&2 &0 \\ 0&0 &2 \end{bmatrix}$

Thus |B| = 24
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