Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Consider the system of linear equations:<br>\(x_1 + 2x_2 + x_3 = 3\)<br>\(2x_1 + 3x_2 + x_3 = 3\)<br>\(3x_1 + 5x_2 + 2x_3 = 1\)<br>The system has</p>
<p>exactly 3 solutions.</p>
<p>a unique solution.</p>
<p>no solution.</p>
<p>infinite number of solutions.</p>
Step-by-Step Solution
Key Concept: Use Gaussian elimination or determinant analysis to check consistency of the augmented matrix. The rank of coefficient matrix A and augmented matrix [A|B] determine whether the system has unique, infinite, or no solutions.
<p><strong>Step 1:</strong> Write the augmented matrix and row reduce:</p><p>$$\left[\begin{array}{ccc|c}1 & 2 & 1 & 3\\2 & 3 & 1 & 3\\3 & 5 & 2 & 1\end{array}\right]$$</p><p><strong>Step 2:</strong> Apply R₂ → R₂ - 2R₁ and R₃ → R₃ - 3R₁:</p><p>$$\left[\begin{array}{ccc|c}1 & 2 & 1 & 3\\0 & -1 & -1 & -3\\0 & -1 & -1 & -8\end{array}\right]$$</p><p><strong>Step 3:</strong> Apply R₃ → R₃ - R₂:</p><p>$$\left[\begin{array}{ccc|c}1 & 2 & 1 & 3\\0 & -1 & -1 & -3\\0 & 0 & 0 & -5\end{array}\right]$$</p><p><strong>Step 4:</strong> The last row represents 0 = -5, which is impossible. Rank(A) = 2 but Rank(A|B) = 3.</p><p>∴ Answer: The system has <strong>no solution (inconsistent)</strong></p>
Correct Answer: C