NIMCET 2011 Question Paper with solutions

Hint: In order to determine the value of $(A+B)^2$ , multiply the both side of the given equation $B=-A^{-1} B A$ with the $A$ . Remember that the multiplication of a matrix with its inverse gives the identity matrix. Now expand the $(A+B)^2$ as $(A+B)(A+B)$ .simplify it further and use the result obtained earlier in this expansion to get the required result. Complete step-by-step answer: We are given two square matrices $A$ and $B$ having relation between as $B=-A^{-1} B A$ Since, $B=-A^{-1} B A$ Let's multiply $A$ on both sides of the equation, we get $A B=-A A^{-1} B A$ As we know when a matrix is multiplied with its inverse matrix, it results into identity matrix. So here $A A^{-1}=I$ $A B=-I B A$ $A B=-B A \; \; \; \; \; \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot (1)$ We have to find the value of $(A+B)^2$, lets rewrite this thing $\Rightarrow (A+B)^2 = (A+B)(A+B)$ $\Rightarrow (A+B)^2 = A \cdot A + A \cdot B + B \cdot A + B \cdot B$ From the equation (1) $A B=-B A$, putting this into the above expression we get $\Rightarrow (A+B)^2 = A^2 - B A + B A + B^2$ $\Rightarrow (A+B)^2 = A^2 + B^2$ Therefore the $(A+B)^2$ is equal to $A^2 + B^2$ .