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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 an identity matrix.
Solution
7A−(I+A)3=7A−[I3+A3+3⋅I2⋅A+3⋅I⋅A2]
=7A−(I+A3+3A+3A2)
=7A−(I+A2⋅A+3A+3A2)
=7A−(I+A⋅A+3A+3A) (∵A2=A)
=7A−(I+A2+6A)
=7A−(I+A+6A)
=7A−(I+7A)
=7A−I−7A
=−I
∴ 7A−(I+A)3=−I
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