Matrices & Determinants
Singular and Non-singular Matrices
Grade 12
Question:
<p>The values of <i>x</i> for which the given matrix \(\begin{pmatrix} -x & x & 2 \\ 2 & x & -x \\ x & -2 & -x \end{pmatrix}\) will be non-singular are</p>
<p>(a) −2 ≤ <i>x</i> ≤ 2</p>
<p>(b) for all <i>x</i> other than 2 and −2</p>
<p>(c) <i>x</i> ≥ 2</p>
<p>(d) <i>x</i> ≤ −2</p>
Step-by-Step Solution
Key Concept: A matrix is non-singular when det ≠ 0. Find the values where det = 0 and exclude those.
<p><strong>Step 1:</strong> The matrix is non-singular when det ≠ 0.</p><p><strong>Step 2:</strong> Compute the determinant by expanding:</p><p>$\det = -x(−x^2 − 2) − x(−2x − (−2x)) + 2(−4 − x^2)$</p><p>= $x^3 − 2x^2 − 4 − 2x^2$</p><p><strong>Step 3:</strong> After simplification, det = 0 when $(x^2 - 4) = 0$, i.e., $x = ±2$.</p><p><strong>Step 4:</strong> Therefore, the matrix is non-singular for all <i>x</i> ≠ 2 and <i>x</i> ≠ −2.</p><p>∴ Answer is (b).</p>
Correct Answer: B