If $A = [\begin{array}{ll} 4 1 \\ 3 2 \end{array}]$ and $I = [\begin{array}{ll} 1 0 \\ 0 1 \end{array}]$ , then $A^{2} - 6 A$ is equal to

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$3 I$

$- 5 I$

$5 I$

none of these

answer is B.

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Detailed Solution

$A^{2} = [\begin{array}{ll} 4 1 \\ 3 2 \end{array}] [\begin{array}{ll} 4 1 \\ 3 2 \end{array}] = [\begin{array}{ll} 19 6 \\ 18 7 \end{array}]$

So, $A^{2} - 6 A = [\begin{array}{ll} 19 6 \\ 18 7 \end{array}] - 6 [\begin{array}{ll} 4 1 \\ 3 2 \end{array}]$

$= [\begin{array}{ll} 19 6 \\ 18 7 \end{array}] - [\begin{array}{rr} 24 6 \\ 18 12 \end{array}] = [\begin{array}{rr} - 5 0 \\ 0 - 5 \end{array}]$

$= - 5 [\begin{array}{ll} 1 0 \\ 0 1 \end{array}] = - 5 I$

$∴$ Option (b) is the correct answer.

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