Question

If $A=\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right],B=\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]$ and (A + B)² = A² + B², values of a and b are
(a) a = 4, b = 1
(b) a = 1, b = 4
(c) a = 0, b = 4
(d) a = 2, b = 4

Answer 1

(b) a = 1, b = 4

$$\begin{gathered} \mathrm{Here}, \\ {\left(A+B\right)}^2=A^2+B^2 \\ \Rightarrow A^2+AB+BA+B^2=A^2+B^2 \\ \Rightarrow AB+BA=O \\ \Rightarrow AB=-BA \\ \Rightarrow \left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right]\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]=-\left[\begin{array}{cc}a& 1\\ b& -1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 2& -1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}a-b& 2\\ 2a-b& 3\end{array}\right]=-\left[\begin{array}{cc}a+2& -a-1\\ b-2& -b+1\end{array}\right] \\ \Rightarrow \left[\begin{array}{cc}a-b& 2\\ 2a-b& 3\end{array}\right]=\left[\begin{array}{cc}-a-2& a+1\\ b+2& b-1\end{array}\right] \\ \mathrm{The}\mathrm{corresponding}\mathrm{elements}\mathrm{of}\mathrm{two}\mathrm{equal}\mathrm{matrices}\mathrm{are}\mathrm{equal}. \\ \Rightarrow a+1=2\mathrm{and}b-1=3 \\ \therefore a=1\mathrm{and}b=4 \end{gathered}$$