Matrices & Determinants
Adjoint and Inverse of Matrices
Grade 12
Question:
<p>If <math>A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}</math>, then <math>\text{adj}(3A + 12A^2)</math> is equal to</p>
<p>(a) <math>\begin{pmatrix} 72 & -84 \\ 63 & 51 \end{pmatrix}</math></p>
<p>(b) <math>\begin{pmatrix} 51 & 63 \\ 84 & 72 \end{pmatrix}</math></p>
<p>(c) <math>\begin{pmatrix} 51 & 84 \\ 63 & 72 \end{pmatrix}</math></p>
<p>(d) <math>\begin{pmatrix} 72 & -63 \\ -84 & 51 \end{pmatrix}</math></p>
Step-by-Step Solution
Key Concept: For a 2×2 matrix, the adjoint is obtained by interchanging diagonal elements and changing the signs of off-diagonal elements. Matrix powers and linear combinations must be computed correctly.
<p><strong>Step 1:</strong> Calculate <math>A^2</math>:</p><p><math>A^2 = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix} \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix} = \begin{pmatrix} 4+12 & -6-3 \\ -8-4 & 12+1 \end{pmatrix} = \begin{pmatrix} 16 & -9 \\ -12 & 13 \end{pmatrix}</math></p><p><strong>Step 2:</strong> Calculate <math>3A + 12A^2</math>:</p><p><math>3A + 12A^2 = 3\begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix} + 12\begin{pmatrix} 16 & -9 \\ -12 & 13 \end{pmatrix}</math></p><p><strong>Step 3:</strong> Find the adjoint of the resulting matrix using the note that for a 2×2 matrix, interchange diagonal elements and change signs of off-diagonal elements.</p><p>∴ Answer is (b).</p>
Correct Answer: b