Given that \(P = \begin{pmatrix} -2 & 1 \\ 3 & 4 \end{pmatrix}\) and \(Q = \begin{pmatrix} 5 & -3 \\ 2 & -1 \end{pmatrix}\), find PQ - QP.
A. \(\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)
B. \(\begin{pmatrix} 27 & 12 \\ 16 & -15 \end{pmatrix}\)
C. \(\begin{pmatrix} -20 & -6 \\ 12 & -8 \end{pmatrix}\)
D. \(\begin{pmatrix} 11 & 12 \\ 30 & -11 \end{pmatrix}\)
Correct Answer: D
Explanation
\(P = \begin{pmatrix} -2 & 1 \\ 3 & 4 \end{pmatrix}; Q = \begin{pmatrix} 5 & -3 \\ 2 & -1 \end{pmatrix}\)
= \(PQ = \begin{pmatrix} -10+2 & 6-1 \\ 15+8 & -9-4 \end{pmatrix}\)
= \(\begin{pmatrix} -8 & 5 \\ 23 & -13 \end{pmatrix}\)
\(QP = \begin{pmatrix} -10-9 & 5-12 \\ -4-3 & 2-4 \end{pmatrix}\)
= \(\begin{pmatrix} -19 & -7 \\ -7 & -2 \end{pmatrix}\)
\(PQ - QP = \begin{pmatrix} -8 & 5 \\ 23 & -13 \end{pmatrix} - \begin{pmatrix} -19 & -7 \\ -7 & -2 \end{pmatrix}\)
= \(\begin{pmatrix} 11 & 12 \\ 30 & -11 \end{pmatrix}\)
