Matrices & Determinants
Inverse of a Matrix
Grade 12
Question:
<p>Which of the following matrices is NOT left inverse of matrix \(\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ 2 & 3 \end{bmatrix}\)?</p>
<p>\(\begin{bmatrix} \frac{1}{2} & \frac{1}{2} & 0 \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 2 & -7 & 3 \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} -\frac{1}{2} & \frac{1}{2} & 0 \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}\)</p>
<p>\(\begin{bmatrix} 0 & 3 & -1 \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}\)</p>
Step-by-Step Solution
Key Concept: A left inverse L of matrix A satisfies LA = I (identity). For a 3×2 matrix A, any left inverse must be 2×3, and we need LA = I₂. Check each option by multiplication to find which does NOT satisfy this condition.
<p><strong>Step 1:</strong> For matrix A = [1 -1; 1 1; 2 3] (3×2), a left inverse L must be 2×3 such that LA = I₂.</p><p><strong>Step 2:</strong> The left inverse must satisfy: L·A = [1 0; 0 1]</p><p><strong>Step 3:</strong> Set up the system. If L = [a b c; d e f], then:</p><p>LA = [a+b+2c, -a+b+3c; d+e+2f, -d+e+3f] = [1 0; 0 1]</p><p><strong>Step 4:</strong> This gives us: a+b+2c=1, -a+b+3c=0, d+e+2f=0, -d+e+3f=1</p><p><strong>Step 5:</strong> From equations 1&2: 2a-c=1 and from 3&4: 2d-f=1. One valid left inverse is L = [1/2 1/2 -1/2; -1/2 1/2 1/2]</p><p><strong>Step 6:</strong> Test each given option by computing LA. The option that does NOT yield I₂ is NOT a left inverse.</p><p>∴ Answer: C</p>
Correct Answer: C