If A=42−11, then (A−2I)(A−3I) is equal to:

Given: A=42−11(A−2I)(A−3I)=42−11−200242−11−3003=22−1−112−1−2=2−24−4−1+1−2+2=0000=O

If A=42−11, then (A−2I)(A−3I) is equal to:

Given: A=42−11(A−2I)(A−3I)=42−11−200242−11−3003=22−1−112−1−2=2−24−4−1+1−2+2=0000=O

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If $A = [\begin{array}{cc} 4 & 2 \\ - 1 & 1 \end{array}], then (A - 2 I) (A - 3 I)$ is equal to:

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answer is D.

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Detailed Solution

Given: $A = [\begin{array}{cc} 4 & 2 \\ - 1 & 1 \end{array}]$
$\begin{array}{l} (A - 2 I) (A - 3 I) = ([\begin{array}{lc} 4 & 2 \\ - 1 & 1 \end{array}] - [\begin{array}{cc} 2 & 0 \\ 0 & 2 \end{array}]) ([\begin{array}{lc} 4 & 2 \\ - 1 & 1 \end{array}] - [\begin{array}{cc} 3 & 0 \\ 0 & 3 \end{array}]) \\ = [\begin{array}{cc} 2 & 2 \\ - 1 & - 1 \end{array}] [\begin{array}{cc} 1 & 2 \\ - 1 & - 2 \end{array}] \\ = [\begin{array}{cc} 2 - 2 & 4 - 4 \\ - 1 + 1 & - 2 + 2 \end{array}] \\ = [\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}] \\ = O \end{array}$