Question
If $A=\left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right], B=\left[\begin{array}{cc}2 & a \\ -1 & b\end{array}\right]$ and $(A+B)^2=A^2+B^2$, find the values of $a$ and $b$.
$\therefore \mathrm{A}^2+\mathrm{AB}+\mathrm{BA}+\mathrm{B}^2=\mathrm{A}^2+\mathrm{B}^2$
∴ AB + BA = 0 ∴ AB = -BA
$\therefore \quad\left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right]\left[\begin{array}{cc}2 & a \\ -1 & b\end{array}\right]=-\left[\begin{array}{cc}2 & a \\ -1 & b\end{array}\right]\left[\begin{array}{cc}1 & 2 \\ -1 & -2\end{array}\right]$
$\therefore \quad\left[\begin{array}{cc}2-2 & a+2 b \\ -2+2 & -a-2 b\end{array}\right]=-\left[\begin{array}{cc}2-a & 4-2 a \\ -1-b & -2-2 b\end{array}\right]$
∴ by equality of matrices, we get – 2 + a = 0 and 1 + b = 0 a = 2 and b = -1 [Note: The question has been modified.]
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