Let A, B and C be three 2 × 2 matrices with real entries

Q. Let A, B and C be three 2 × 2 matrices with real entries such that B = (I + A)⁻¹ and A + C = I.

If BC = [ 1−5 −12 ] and

CB [ x₁ x₂ ] = [ 12 −6 ] , then x₁ + x₂ is

(A) 4

(B) 2

(C) 0

(D) −2

Correct Answer: 0

Explanation

B = (I + A)⁻¹ and A + C = I

From A + C = I, we get

C = I − A

Now,

CB = (I − A)B = B − AB

Since,

(I + A)B = I

⇒ AB = I − B

Substitute:

CB = B − (I − B) = 2B − I

Thus CB is fully known.

Solving CB [ x₁ x₂ ] = [ 12 −6 ] gives

x₁ + x₂ = 0

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