Question

if $A = \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix} , B =\begin{bmatrix}x&1\\ y&-1\end{bmatrix}$ and $\left(A + B\right)^{2} =A^{2} + B^{2},$ then $x + y$ =

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

Answer: (D)

$ \left(A + B\right)^{2 }= A^{2 }+ B^{2} \Rightarrow AB+ BA= 0$

$\Rightarrow \begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}\begin{bmatrix}x&1\\ y&-1\end{bmatrix}+\begin{bmatrix}x&1\\ y&-1\end{bmatrix}\begin{bmatrix}1&-1\\ 2&-1\end{bmatrix}=0$

$\Rightarrow \begin{bmatrix}x-y&2\\ 2x-y&3\end{bmatrix}+\begin{bmatrix}x+2&-x-1\\ y-2&-y+1\end{bmatrix}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix}$

$\Rightarrow 2x-y+2=0....\left(i\right), -x+1=0....\left(ii\right)$

$2x-2=0....\left(iii\right) and-y+4 =0\backslash,\backslash,....\left(iv\right)$

From $\left(ii\right), x = 1$ and from $\left(iv\right), y = 4$

Now, $x + y = 1 + 4 = 5 $