Matrices & Determinants
Adjoint of Matrix
Grade 12
Question:
<p>If <span class="math">\(A = \begin{pmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{pmatrix}\)</span> and B is the adjoint of A, find the value of <span class="math">\(|AB + 2I|\)</span>, where I is the identity matrix of order 3.</p>
Step-by-Step Solution
Key Concept: Use the property that if B = adj A, then AB = |A|I, and apply the determinant formula |AB + kI| = (|A| + k)^n
<p><strong>Step 1:</strong> Calculate <span class="math">$|A|$</span></p><p><span class="math">$|A| = 3(-3 + 4) + 3(2 - 0) + 4(-2 - 0) = 3(1) + 3(2) + 4(-2) = 3 + 6 - 8 = 1$</span></p><p><strong>Step 2:</strong> Apply Property 8</p><p>Since B is the adjoint of A, we have <span class="math">$AB = A(\text{adj } A) = |A| I_n$</span></p><p><strong>Step 3:</strong> Calculate <span class="math">$|AB + 2I|$</span></p><p><span class="math">$|AB + 2I| = ||A|I + 2I| = |(|A| + 2)I| = (|A| + 2)^3 = (1 + 2)^3 = 3^3 = 27$</span></p><p>∴ Answer is <strong>27</strong>.</p>
Correct Answer: 27