If  $\mathrm{A}=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]$ ,$\mathrm{B}=\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& -1\end{array}\right]$ and ${\left(\mathrm{A}+\mathrm{B}\right)}^2={\mathrm{A}}^2+{\mathrm{B}}^2$ , the value of a + b is

$$\begin{gathered} {\left(A+B\right)}^2=A^2+B^2\Rightarrow AB+BA=0 \\ \Rightarrow \left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& -1\end{array}\right]+\left[\begin{array}{cc}\mathrm{a}& 1\\ \mathrm{b}& -1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]=0 \\ \Rightarrow \left[\begin{array}{cc}a-b& 2\\ 2a-b& 3\end{array}\right]+\left[\begin{array}{cc}a+2& -a-1\\ b-2& -b+1\end{array}\right]=0\Rightarrow a=1,b=4\Rightarrow a+b=5 \end{gathered}$$