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Involutory Matrix

If A is an id...

Question

If $A$ is an idempotent matrix and $(2 I + A)^{4} = a I + b A, (I is unit matrix), a, b \in R .$ Then $(a + b) =$

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Solution

For $A$ to be an idempotent matrix : $A^{n} = A, n \geq 2, n \in N$
Now,
$(2 I + A)^{4} = C_{0} (2 I)^{4} + C_{1} (2 I)^{3} A + C_{2} (2 I)^{2} A^{2} + C_{3} (2 I) A^{3} + C_{4} A^{4} = 2^{4} I + (8 \cdot C_{1} + 4 \cdot C_{2} + 2 \cdot C_{3} + C_{4}) A = 16 I + 65 A$
So, $a + b = 81$

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Involutory Matrix

Standard IX Mathematics

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