Matrices & Determinants
Singular Matrices and Determinants
Grade 12
Question:
<p>The number of values of <i>x</i> in the closed interval [−4, −1], for which the matrix \(\begin{pmatrix} 3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2 \end{pmatrix}\) is singular, is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) 2</p>
<p>(d) 3</p>
Step-by-Step Solution
Key Concept: Find all values where det = 0, then count how many solutions lie in the given closed interval.
<p><strong>Step 1:</strong> The matrix is singular when det = 0.</p><p><strong>Step 2:</strong> Compute the determinant and set it equal to zero, yielding a polynomial equation in <i>x</i>.</p><p><strong>Step 3:</strong> Solve the polynomial to find all roots.</p><p><strong>Step 4:</strong> Check which roots lie in the interval [−4, −1]. Only 1 root falls within this interval.</p><p>∴ Answer is (b).</p>
Correct Answer: B