<p>Solve for <i>x</i> if the determinant \[\begin{vmatrix} <i>t</i>-1 & 3<i>t</i>+1 & 2<i>t</i> \\ <i>t</i>-1 & 4<i>t</i>-2 & <i>t</i>+3 \\ 2 & 3<i>t</i>+1 & 3(<i>t</i>-1) \end{vmatrix} = 0\]</p>
Step-by-Step Solution
Key Concept: Using row operations to simplify the determinant before expansion makes the problem tractable. Factoring out common terms reveals the solution quickly.
<p><strong>Step 1:</strong> Apply row operations <i>R</i><sub>2</sub> → <i>R</i><sub>2</sub> − <i>R</i><sub>1</sub> and <i>R</i><sub>3</sub> → <i>R</i><sub>3</sub> − <i>R</i><sub>1</sub>:</p><p>$$\begin{vmatrix} <i>t</i>-1 & 3<i>t</i>+1 & 2<i>t</i> \\ 0 & <i>t</i>-3 & 3-<i>t</i> \\ 3-<i>t</i> & 0 & <i>t</i>-3 \end{vmatrix}$$</p><p><strong>Step 2:</strong> Factor out (3 − <i>t</i>) and (−1) from appropriate rows/columns.</p><p><strong>Step 3:</strong> Expand the determinant to get:</p><p>(<i>t</i> − 3)<sup>2</sup>[(<i>t</i> − 1)(1) − (3<i>t</i> + 1)(−1) + 2<i>t</i>(1)] = 0</p><p><strong>Step 4:</strong> Simplify: (<i>t</i> − 3)<sup>2</sup>[<i>t</i> − 1 + 3<i>t</i> + 1 + 2<i>t</i>] = 0</p><p>(<i>t</i> − 3)<sup>3</sup>[6<i>t</i>] = 0</p><p>∴ <i>t</i> = 0 or <i>t</i> = 3</p>
Correct Answer: 0 or 3