If $A = [\begin{matrix} 1 & 1 & - 2 \\ 0 & 1 & 4 \\ 2 & 3 & 1 \end{matrix}] a n d A^{3} - 3 A^{2} - 5 A = k I t h e n k = . . . . . . . . . . . . .$

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$3$

$2$

$1$

$4$

answer is B.

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Detailed Solution

$A^{2} = [\begin{matrix} 1 & 1 & - 2 \\ 0 & 1 & 4 \\ 2 & 3 & 1 \end{matrix}] [\begin{matrix} 1 & 1 & - 2 \\ 0 & 1 & 4 \\ 2 & 3 & 1 \end{matrix}] = [\begin{matrix} 1 + 0 - 4 & 1 + 1 - 6 & - 2 + 4 - 2 \\ 0 + 0 + 8 & 0 + 1 + 12 & 0 + 4 + 4 \\ 2 + 0 + 2 & 2 + 3 + 3 & - 4 + 12 + 1 \end{matrix}] = [\begin{matrix} - 3 & - 4 & 0 \\ 8 & 13 & 8 \\ 4 & 8 & 9 \end{matrix}]$

$A^{3} = A A^{2} = [\begin{matrix} 1 & 1 & - 2 \\ 0 & 1 & 4 \\ 2 & 3 & 1 \end{matrix}] [\begin{matrix} 1 & 1 & - 2 \\ 8 & 13 & 8 \\ 4 & 8 & 9 \end{matrix}] = [\begin{matrix} - 3 + 8 - 8 & - 4 + 13 - 16 & 0 + 8 - 18 \\ 0 + 8 + 16 & 0 + 13 + 32 & 0 + 8 + 36 \\ - 6 + 24 + 4 & - 8 + 39 + 8 & 0 + 24 + 9 \end{matrix}] = [\begin{matrix} - 3 & - 7 & - 10 \\ 24 & 45 & 44 \\ 22 & 39 & 33 \end{matrix}]$

$A^{3} - 3 A^{2} - 5 A - I = [\begin{matrix} - 3 & - 7 & - 10 \\ 24 & 45 & 44 \\ 22 & 39 & 33 \end{matrix}] - 3 [\begin{matrix} - 3 & - 4 & 0 \\ 8 & 13 & 8 \\ 4 & 8 & 9 \end{matrix}] = 5 [\begin{matrix} 1 & 1 & - 2 \\ 0 & 1 & 4 \\ 2 & 3 & 1 \end{matrix}] - [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

= $[\begin{matrix} - 3 + 9 - 5 - 1 & - 7 + 12 - 5 - 0 & - 10 - 0 + 10 - 0 \\ 24 - 24 - 0 - 0 & 45 - 39 - 5 - 1 & 44 - 24 - 20 - 0 \\ 22 - 12 - 10 - 0 & 39 - 24 - 15 - 0 & 33 - 27 - 5 - 1 \end{matrix}] = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] = O$