NIMCET 2023 — Mathematics PYQ

To find the value of $(A+B{)}^2$, we need to expand the expression using the basic algebraic laws of matrix multiplication.

Step 1: Expand $(A+B{)}^2$

Since matrix multiplication is generally not commutative ($AB\ne BA$), we expand it carefully:

$$(A+B{)}^2=(A+B)(A+B)$$

$$(A+B{)}^2=A(A+B)+B(A+B)$$

$$(A+B{)}^2=A^2+AB+BA+B^2\text{— (Equation 1)}$$

Step 2: Simplify the given condition

We are given the relation:

$$B=-A^{-1}BA$$

To remove the inverse matrix $A^{-1}$, pre-multiply both sides of the equation by matrix $A$:

$$A\cdot B=A\cdot (-A^{-1}BA)$$

$$AB=-(A\cdot A^{-1})BA$$

Since $A\cdot A^{-1}=I$ (the Identity Matrix) and $I\cdot B=B$:

$$AB=-I\cdot BA$$

$$AB=-BA$$

Step 3: Rearrange the relation

From $AB=-BA$, we can add $BA$ to both sides to find:

$$AB+BA=0$$

Alternatively, this shows that the two terms are negatives of each other, meaning they cancel each other out when added.

Step 4: Substitute this back into Equation 1

Now, substitute $AB+BA=0$ into our original expansion from Step 1:

$$(A+B{)}^2=A^2+(AB+BA)+B^2$$

$$(A+B{)}^2=A^2+0+B^2$$

$$(A+B{)}^2=A^2+B^2$$

Correct Answer: B) $A^2+B^2$