Find the value of a and b

D e a r s t u d e n t, A = [\begin{matrix} 1 & - 1 \\ 2 & - 1 \end{matrix}]; B = [\begin{matrix} a & 1 \\ b & - 1 \end{matrix}] A + B = [\begin{matrix} 1 + a & 0 \\ 2 + b & - 2 \end{matrix}] A^{2} = [\begin{matrix} 1 & - 1 \\ 2 & - 1 \end{matrix}] \times [\begin{matrix} 1 & - 1 \\ 2 & - 1 \end{matrix}] = [\begin{matrix} 1 - 2 & - 1 + 1 \\ 2 - 2 & - 2 + 1 \end{matrix}] = [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] B^{2} = [\begin{matrix} a & 1 \\ b & - 1 \end{matrix}] \times [\begin{matrix} a & 1 \\ b & - 1 \end{matrix}] = [\begin{matrix} a^{2} + b & a - 1 \\ a b - b & b + 1 \end{matrix}] {(A + B)}^{2} = [\begin{matrix} 1 + a & 0 \\ 2 + b & - 2 \end{matrix}] \times [\begin{matrix} 1 + a & 0 \\ 2 + b & - 2 \end{matrix}] = [\begin{matrix} {(1 + a)}^{2} & 0 \\ (2 + b) (1 + a) - 2 (2 + b) & 4 \end{matrix}] G i v e n, A^{2} + B^{2} = {(A + B)}^{2} [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] + [\begin{matrix} a^{2} + b & a - 1 \\ a b - b & b + 1 \end{matrix}] = [\begin{matrix} {(1 + a)}^{2} & 0 \\ (2 + b) (1 + a) - 2 (2 + b) & 4 \end{matrix}] [\begin{matrix} a^{2} + b - 1 & a - 1 \\ a b - b & b \end{matrix}] = [\begin{matrix} {(1 + a)}^{2} & 0 \\ (2 + b) (1 + a) - 2 (2 + b) & 4 \end{matrix}] S o, a - 1 = 0 \Rightarrow a = 1 b = 4 R e g a r d s

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