Matrices & Determinants
Matrix Operations
Grade 12
Question:
<p>If <m>A = \begin{pmatrix} 4 & 1 \\ 7 & 2 \end{pmatrix}</m> and <m>B = \begin{pmatrix} 2 & -1 \\ 7 & 4 \end{pmatrix}</m>, then <m>B^T A^T</m> is</p>
<p>(a) <m>\begin{pmatrix} 4 & 1 \\ -4 & 0 \end{pmatrix}</m></p>
<p>(b) <m>\begin{pmatrix} 3 & 1 \\ -4 & -2 \end{pmatrix}</m></p>
<p>(c) <m>\begin{pmatrix} 5 & 2 \\ -3 & -8 \end{pmatrix}</m></p>
<p>(d) <m>\begin{pmatrix} 5 & 2 \\ 8 & -3 \end{pmatrix}</m></p>
Step-by-Step Solution
Key Concept: Compute the transposes of both matrices, then multiply B^T and A^T in the correct order.
<p><strong>Step 1:</strong> Calculate <m>A^T = \begin{pmatrix} 4 & 7 \\ 1 & 2 \end{pmatrix}</m>.</p><p><strong>Step 2:</strong> Calculate <m>B^T = \begin{pmatrix} 2 & 7 \\ -1 & 4 \end{pmatrix}</m>.</p><p><strong>Step 3:</strong> Compute <m>B^T A^T = \begin{pmatrix} 2 & 7 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} 4 & 7 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 4 & 1 \\ -4 & 0 \end{pmatrix}</m>.</p><p>∴ Answer is (a).</p>
Correct Answer: A