Matrices & Determinants
Trace of a Matrix
Grade 12
Question:
<p>Elements of a matrix <i>A</i> of order <span>10 × 10</span> are defined as <i>a</i><sub>ij</sub> = <i>ω</i><sup>i+j</sup> (where <i>ω</i> is cube root of unity), then trace(<i>A</i>) of the matrix is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 3</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the property that for cube root of unity ω, we have 1 + ω + ω² = 0. Group the powers of ω in the trace sum accordingly.
<p><strong>Solution:</strong></p><p>tr(<i>A</i>) = $\sum_{i=j=1}^{10} a_{ij} = \sum_{i=j=1}^{10} \omega^{i+j} = \sum_{i=1}^{10} \omega^{2i}$</p><p>= $\omega^2 + \omega^4 + \omega^6 + \omega^8 + \cdots + \omega^{20}$</p><p>= $(\omega^2 + \omega + 1) + (\omega^2 + \omega + 1) + (\omega^2 + \omega + 1) + \omega^2$</p><p>= $0 + 0 + 0 + \omega^2 = \omega^2$</p><p>∴ Answer is (d) None of these.</p>
Correct Answer: D