If A is a square matrix, such that A^{2}=A, then write the value of 7A−(I+A)^{3}, where I is an identity matrix.

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#### Solution

7A−(I+A)^{3}=7A−[I^{3}+A^{3}+3⋅I^{2}⋅A+3⋅I⋅A^{2}]

=7A−(I+A^{3}+3A+3A^{2})

=7A−(I+A^{2}⋅A+3A+3A^{2})

=7A−(I+A⋅A+3A+3A) (∵A^{2}=A)

=7A−(I+A^{2}+6A)

=7A−(I+A+6A)

=7A−(I+7A)

=7A−I−7A

=−I

∴ 7A−(I+A)^{3}=−I

Concept: Types of Matrices

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