You visited us 1 times! Enjoying our articles? Unlock Full Access!

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 $(a + b) =$

Open in App

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, $a + b = 81$

Suggest Corrections

Similar questions

A

(I + A)^{n} = a I + b A, (I is unit matrix), a, b \in R .

(a + b) =

A

A^{2} = A

A

B

A B = B A

A = I - B

A = I - B

A

A^{2} = A

B = I - A

Join BYJU'S Learning Program

Explore more

Join BYJU'S Learning Program