Discussion
If \(A = \begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix}\), \(B = \begin{bmatrix} a & 1 \\ b & -1 \end{bmatrix}\) and \((A + B)^{2} = A^{2} + B^{2}\), then the values of a and b are:
Answer: B
The values are a = 1, b = 4.
Expanding, \((A + B)^{2} = A^{2} + AB + BA + B^{2}\), so the condition holds exactly when \(AB + BA = 0\). Matrix products do not commute, so AB + BA is not 2AB.
\[ \begin{aligned} AB &= \begin{bmatrix} a - b & 2 \\ 2a - b & 3 \end{bmatrix} \\ BA &= \begin{bmatrix} a + 2 & -a - 1 \\ b - 2 & 1 - b \end{bmatrix} \end{aligned} \]
\[ AB + BA = \begin{bmatrix} 2a - b + 2 & 1 - a \\ 2a - 2 & 4 - b \end{bmatrix} \]
The top-right entry gives a = 1 and the bottom-right gives b = 4; then 2a - b + 2 = 0 and 2a - 2 = 0 check. The other pairs fail at once: a = 4 or a = 0 or a = 2 leaves 1 - a non-zero.
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