NIMCET 2011 — Mathematics PYQ

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Question

NIMCET 2011 — Mathematics PYQ

NIMCET | Mathematics | 2011

If $A$ and $B$ are two square matrices such that $B = -A^{-1}BA$ , then $(A + B)^2$ will be equal to:

Choose the correct answer:

Explanation

1. Simplify the given condition:

Given:

B = -A^{-1}BA

Multiply both sides by $A$ from the left:

AB = A(-A^{-1}BA)

AB = -(AA^{-1})BA

Since $AA^{-1} = I$ (Identity matrix):

AB = -IBA

AB = -BA

2. Rearrange the relation:

AB + BA = 0

3. Expand the required expression:

We need to find the value of $(A + B)^2$ . For matrices, the expansion is:

(A + B)^2 = (A + B)(A + B)

(A + B)^2 = A^2 + AB + BA + B^2

4. Substitute the condition from Step 2:

Substitute $AB + BA = 0$ into the expansion:

(A + B)^2 = A^2 + (AB + BA) + B^2

(A + B)^2 = A^2 + 0 + B^2

(A + B)^2 = A^2 + B^2

Conclusion:

The expression simplifies to $A^2 + B^2$ .

The correct option is (c).