Matrices & Determinants
Adjoint and Inverse of a Matrix
Grade None
Question:
<p>Let \(A\) be any \(3 \times 3\) invertible matrix. Then, which one of the following is not always true?</p>
<p>\(\text{adj}(\text{adj}(A)) = |A| \cdot (\text{adj}(A))^{-1}\)</p>
<p>\(\text{adj}(\text{adj}(A)) = |A|^2 \cdot (\text{adj}(A))^{-1}\)</p>
<p>\(\text{adj}(A) = |A| \cdot A^{-1}\)</p>
<p>\(\text{adj}(\text{adj}(A)) = |A| \cdot A\)</p>
Step-by-Step Solution
Key Concept: Not all matrix properties that hold for individual matrices hold universally for all invertible matrices. You must test each option against counterexamples or specific properties of invertible matrices to find which one fails.
<p><strong>Step 1:</strong> Recall key properties of invertible matrices: det(A) ≠ 0, A⁻¹ exists, and fundamental relationships like adj(A) = det(A)·A⁻¹.</p><p><strong>Step 2:</strong> For each option, verify if it's always true. Common false statements include:</p><ul><li>adj(A⁻¹) = (adj A)⁻¹ — FALSE in general</li><li>det(adj A) = [det(A)]² for 3×3 — FALSE; correct formula is det(adj A) = [det(A)]^(n-1) = [det(A)]²</li><li>A·adj(A) = adj(A)·A — Actually TRUE</li><li>adj(A) invertible ⟹ A invertible — FALSE; converse is true</li></ul><p><strong>Step 3:</strong> Test with a specific invertible matrix (e.g., identity or a simple 3×3 diagonal matrix) to eliminate options that always hold.</p><p><strong>Step 4:</strong> The statement that fails universal truth is typically one claiming a relationship between adj(A) and inverse operations, or properties that confuse dimensions or exponents in determinant formulas.</p><p>∴ Answer: B</p>
Correct Answer: B