Matrices & Determinants
Adjoint and Inverse
Grade 12
Question:
<p>If \(A\) is an invertible matrix, then \((\text{adj } A)^{-1}\) is equal to</p>
<p>(1) \(\text{adj}(A^{-1})\)</p>
<p>(2) \(\dfrac{A}{\det A}\)</p>
<p>(3) \(A\)</p>
<p>(4) \((\det A) \cdot A\)</p>
Step-by-Step Solution
Key Concept: Use the fundamental property adj(A) · A = |A|·I to express (adj A)⁻¹ in terms of A and |A|. The inverse of adj(A) can be found by manipulating this relationship directly without computing adj(A) explicitly.
<p><strong>Step 1:</strong> Recall the fundamental property: <strong>adj(A) · A = |A| · I</strong></p><p><strong>Step 2:</strong> Multiply both sides by A⁻¹ from the right:</p><p>adj(A) · A · A⁻¹ = |A| · I · A⁻¹</p><p>adj(A) · I = |A| · A⁻¹</p><p><strong>Step 3:</strong> Therefore: <strong>adj(A) = |A| · A⁻¹</strong></p><p><strong>Step 4:</strong> Taking inverse of both sides:</p><p>(adj A)⁻¹ = (|A| · A⁻¹)⁻¹ = |A|⁻¹ · (A⁻¹)⁻¹ = <strong>A/|A|</strong></p><p><strong>Step 5:</strong> Alternatively, this equals <strong>(1/|A|) · A</strong> or <strong>adj(A⁻¹)</strong></p><p>∴ Answer: <strong>(adj A)⁻¹ = A/|A|</strong> (or equivalently <strong>adj(A⁻¹)</strong>)</p>
Correct Answer: A