Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Consider the system of linear equations:</p><p>\(x_1 + 2x_2 + x_3 = 3\)</p><p>\(2x_1 + 3x_2 + x_3 = 3\)</p><p>\(3x_1 + 5x_2 + 2x_3 = 1\)</p><p>The system has</p>
<p>(1) no solution.</p>
<p>(2) infinite number of solutions.</p>
<p>(3) exactly three solutions.</p>
<p>(4) a unique solution.</p>
Step-by-Step Solution
Key Concept: Check if the coefficient matrix and augmented matrix have the same rank using row reduction. If rank(A) ≠ rank(A|B), the system is inconsistent; if they're equal but less than n variables, infinitely many solutions exist.
<p><strong>Step 1:</strong> Write the augmented matrix [A|B]:</p><p>$$\begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 2 & 3 & 1 & | & 3 \\ 3 & 5 & 2 & | & 1 \end{bmatrix}$$</p><p><strong>Step 2:</strong> Row reduce. R₂ → R₂ - 2R₁, R₃ → R₃ - 3R₁:</p><p>$$\begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 0 & -1 & -1 & | & -3 \\ 0 & -1 & -1 & | & -8 \end{bmatrix}$$</p><p><strong>Step 3:</strong> R₃ → R₃ - R₂:</p><p>$$\begin{bmatrix} 1 & 2 & 1 & | & 3 \\ 0 & -1 & -1 & | & -3 \\ 0 & 0 & 0 & | & -5 \end{bmatrix}$$</p><p><strong>Step 4:</strong> The last row gives 0 = -5, which is impossible. rank(A) = 2 but rank(A|B) = 3.</p><p>∴ <strong>Answer: No solution (Inconsistent system)</strong></p>
Correct Answer: A