<p>In which of the following type of matrix inverse does not exist always?</p>
Step-by-Step Solution
Key Concept: A matrix inverse exists if and only if the determinant is non-zero (|A| ≠ 0). Singular matrices (where det(A) = 0) never have inverses, and certain matrix types are always singular or frequently singular.
<p><strong>Step 1:</strong> Recall that a square matrix A is invertible if and only if det(A) ≠ 0.</p><p><strong>Step 2:</strong> Examine common matrix types:</p><ul><li><strong>Symmetric matrices:</strong> Can be singular or non-singular (e.g., [[1,0],[0,0]] is singular)</li><li><strong>Skew-symmetric odd-order matrices:</strong> ALWAYS singular (det(A) = -det(A) ⟹ det(A) = 0)</li><li><strong>Orthogonal matrices:</strong> Always invertible (det(A) = ±1)</li><li><strong>Idempotent matrices:</strong> Can be singular (e.g., projection matrices)</li></ul><p><strong>Step 3:</strong> The critical insight is that <strong>skew-symmetric matrices of odd order</strong> always have determinant zero, making their inverse non-existent.</p><p><strong>Proof for odd skew-symmetric:</strong> If A is skew-symmetric, A<sup>T</sup> = -A. Taking determinant: det(A<sup>T</sup>) = det(-A) ⟹ det(A) = (-1)<sup>n</sup>det(A). For odd n: det(A) = -det(A) ⟹ det(A) = 0.</p><p>∴ Answer: A (likely referring to skew-symmetric matrices of odd order, or singular/non-square matrices depending on options provided)</p>
Correct Answer: A