Matrices & Determinants
Diagonal and Scalar Matrices
Grade 12
Question:
<p>If <i>A</i> is a diagonal matrix of order 3 × 3 is commutative with every square matrix of order 3 × 3 under multiplication and trace(<i>A</i>) = 12, then</p>
<p>(a) |<i>A</i>| = 64</p>
<p>(b) |<i>A</i>| = 16</p>
<p>(c) |<i>A</i>| = 12</p>
<p>(d) |<i>A</i>| = 4</p>
Step-by-Step Solution
Key Concept: A diagonal matrix commutes with every square matrix if and only if it is a scalar matrix. Use the trace condition to find the diagonal elements.
<p><strong>Solution:</strong></p><p>A diagonal matrix commutes with every square matrix if and only if it is a scalar matrix (all diagonal elements are equal).</p><p>Let the diagonal elements be <i>k</i>, <i>k</i>, <i>k</i>.</p><p>Since trace(<i>A</i>) = 12, we have:</p><p><i>k</i> + <i>k</i> + <i>k</i> = 12</p><p>3<i>k</i> = 12</p><p><i>k</i> = 4</p><p>Therefore, <i>A</i> = <begin>pmatrix</begin>4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4<end>pmatrix</end></p><p>|<i>A</i>| = 4 · 4 · 4 = 64</p><p>∴ Answer is <strong>(a) |<i>A</i>| = 64</strong></p>
Correct Answer: A