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Question
If A is a square matrix such that A2 = A, then write the value of 7A − (I + A)3, where I is the identity matrix.
Solution
\[7A - \left( I + A \right)^3 = 7A - \left( I^3 + A^3 + 3 A^2 I + 3A I^2 \right)\]
\[ = 7A - \left( I + A . A^2 + 3 A^2 + 3A \right)\]
\[ = 7A - \left( I + A . A + 3A + 3A \right) \left( \because A^2 = A \right)\]
\[ = 7A - \left( I + A^2 + 6A \right)\]
\[ = 7A - \left( I + A + 6A \right) \left( \because A^2 = A \right)\]
\[ = 7A - \left( I + 7A \right)\]
\[ = 7A - I - 7A\]
\[ = - I\]
Hence, the value of 7A − (I + A)3 is −I.
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