Discussion
\([\mathbf{P}] = \begin{bmatrix} 2 & 3 & 7 \\ 4 & 1 & 5 \end{bmatrix}\) and \([\mathbf{Q}] = \begin{bmatrix} 8 & 11 & 9 \\ 12 & 6 & 13 \end{bmatrix}\) are two matrices. The matrix \([\mathbf{P}]^{T}[\mathbf{Q}]\) is __________.
Answer: B
The product is \(\begin{bmatrix} 64 & 46 & 70 \\ 36 & 39 & 40 \\ 116 & 107 & 128 \end{bmatrix}\).
\([\mathbf{P}]\) is 2 × 3, so its transpose is 3 × 2, and a 3 × 2 matrix times the 2 × 3 matrix \([\mathbf{Q}]\) gives a 3 × 3 matrix. The rows of \([\mathbf{P}]^{T}\) are (2, 4), (3, 1) and (7, 5); multiply each by the columns (8, 12), (11, 6) and (9, 13) of \([\mathbf{Q}]\):
\[ \begin{aligned} \text{row 1} &: 16 + 48,\ 22 + 24,\ 18 + 52 \\ &= 64,\ 46,\ 70 \\ \text{row 2} &: 24 + 12,\ 33 + 6,\ 27 + 13 \\ &= 36,\ 39,\ 40 \\ \text{row 3} &: 56 + 60,\ 77 + 30,\ 63 + 65 \\ &= 116,\ 107,\ 128 \end{aligned} \]
The first option is the transpose of the correct answer, \(([\mathbf{P}]^{T}[\mathbf{Q}])^{T} = [\mathbf{Q}]^{T}[\mathbf{P}]\). The 2 × 2 options cannot be right whatever their entries, because a 3 × 2 matrix times a 2 × 3 matrix is always 3 × 3.
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