Matrices & Determinants
Inverse of a Matrix
Grade 12
Question:
<p>For which of the following matrices, the number of left inverses is greater than the number of right inverses?</p>
<p>\(\begin{bmatrix} 1 & 2 & 4 \\ -3 & 2 & 1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 3 & 2 & 1 \\ 3 & 2 & 1 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 1 & 4 \\ 2 & -3 \\ 5 & 4 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 3 & 3 \\ 1 & 1 \\ 4 & 4 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: A matrix has left inverses iff it has full row rank (rank = m for m×n matrix), and right inverses iff it has full column rank (rank = n). For m > n, full row rank is achievable but full column rank is impossible, making left inverses possible but right inverses impossible.
<p><strong>Key Insight:</strong> For an m×n matrix A:</p><p>• <strong>Left inverse</strong> exists ⟺ rank(A) = m (full row rank) ⟺ m ≤ n</p><p>• <strong>Right inverse</strong> exists ⟺ rank(A) = n (full column rank) ⟺ m ≥ n</p><p><strong>Analysis:</strong></p><p><strong>Case 1:</strong> m < n (wide matrix) → Can have right inverses but not left inverses</p><p><strong>Case 2:</strong> m = n (square, full rank) → Has exactly one left and one right inverse (they're equal)</p><p><strong>Case 3:</strong> m > n (tall matrix) → Can have infinitely many left inverses but NO right inverses</p><p><strong>Conclusion:</strong> A tall matrix (more rows than columns, with full row rank) has infinitely many left inverses and zero right inverses.</p><p>∴ Answer: <strong>C</strong> (the tall/rectangular matrix with full row rank)</p>
Correct Answer: C