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Question
If A is a square matrix such that A2 = A, then (I + A)3 − 7A is equal to
Options
A
I-A
I
3A
Solution
I
\[Here, \]
\[ A^2 = A . . . \left( 1 \right)\]
\[ A^3 = A^2 A\]
\[ = A^2 \left[ \text{From eq }. \left( 1 \right) \right] \]
\[ = A \]
\[ \therefore A^3 = A . . . \left( 2 \right)\]
\[\text{We know that} \left( I + A \right)^3 = I^3 + 3 \left( I \right)^2 A + 3\left( I \right) A^2 + A^3 \]
\[ \Rightarrow \left( I + A \right)^3 = I + 3A + 3A + A \left[ \text{From eqs }. \left( 1 \right) and \left( 2 \right) \right] \]
\[ \Rightarrow \left( I + A \right)^3 = I + 7A\]
\[ \Rightarrow \left( I + A \right)^3 - 7A = I\]
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