Matrices & Determinants
Adjoint of a Matrix
Grade 12
Question:
<p>If \(A = [a_{ij}]_{4 \times 4}\), such that \(a_{ij} = \begin{cases} 2, & \text{when } i = j \\ 0, & \text{when } i \neq j \end{cases}\), then \(\left\{ \dfrac{\det(\text{adj}(\text{adj } A))}{7} \right\}\) is (where \(\{\cdot\}\) represents fractional part function)</p>
<p>1/7</p>
<p>2/7</p>
<p>3/7</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: For a diagonal matrix A of order n with diagonal entries, adj(adj A) = (det A)^(n-2) · A. Here A = 2I₄, so det A = 16, and det(adj(adj A)) = (det A)^(n-1) = 16³. The fractional part function extracts the decimal portion after division.
<p><strong>Step 1:</strong> Identify matrix A. Since a_{ij} = 2 when i = j and 0 otherwise, A = 2I₄ (scalar multiple of identity).</p><p><strong>Step 2:</strong> Calculate det(A) = 2⁴ = 16.</p><p><strong>Step 3:</strong> Apply the formula: det(adj(adj A)) = (det A)^((n-1)) where n = 4. Thus det(adj(adj A)) = 16³ = 4096.</p><p><strong>Step 4:</strong> Divide by 7: 4096/7 = 585 + 1/7 = 585.142857...</p><p><strong>Step 5:</strong> Extract fractional part: {4096/7} = 1/7 ≈ 0.1429.</p><p>∴ Answer: C</p>
Correct Answer: C