Answered

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Q. If A is a square matrix such that A² = A, then write the value of 7 A −(I + A)³, where I is an identity matrix.​

Sagot :

Since [tex]A^2=A[/tex], we have

[tex](I + A)^2 = I^2 + IA + AI + A^2 = I + A + A + A = I + 3A[/tex]

[tex](I + A)^3 = (I + A) (I + 3A) = I^2 + 3IA + AI + 3A^2 = I + 3A + A +  3A = I + 7A[/tex]

Then the target expression is

[tex]7A - (I + A)^3 = 7A - (I + 7A) = \boxed{-I}[/tex]

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