Matrices & Determinants
Rank of a matrix
Grade 12
Question:
<p>The rank of the matrix \(\begin{bmatrix} 1 & 2 & 3 \\ \lambda & 2 & 4 \\ 2 & -3 & 1 \end{bmatrix}\) is 3 if</p>
<p>(a) \(\lambda = \dfrac{18}{11}\)</p>
<p>(b) \(\lambda = \dfrac{18}{11}\)</p>
<p>(c) \(\lambda = \dfrac{-18}{11}\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: A matrix has rank 3 (full rank for a 3×3 matrix) if and only if its determinant is non-zero. Calculate det(A) and find the condition on λ that makes it non-zero.
<p><strong>Step 1:</strong> For a 3×3 matrix to have rank 3, its determinant must be non-zero.</p><p><strong>Step 2:</strong> Calculate det(A) using cofactor expansion along the first row:</p><p>det(A) = 1(2·1 - 4·(-3)) - 2(λ·1 - 4·2) + 3(λ·(-3) - 2·2)</p><p>= 1(2 + 12) - 2(λ - 8) + 3(-3λ - 4)</p><p>= 14 - 2λ + 16 - 9λ - 12</p><p>= 18 - 11λ</p><p><strong>Step 3:</strong> For rank to be 3, we need det(A) ≠ 0:</p><p>18 - 11λ ≠ 0</p><p>λ ≠ 18/11</p><p>∴ Answer: D (λ ≠ 18/11)</p>
Correct Answer: D