Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Given the system of linear equations<br>\(2x + 2y + 3z = a\)<br>\(3x - y + 5z = b\)<br>\(x - 3y + 2z = c\)<br>where \(a, b, c\) are non-zero real numbers, the system has more than one solution if</p>
<p>\(b - c + a = 0\)</p>
<p>\(a + b + c = 0\)</p>
<p>\(b + c - a = 0\)</p>
<p>\(c - b + a = b - a - c\)</p>
Step-by-Step Solution
Key Concept: A system of linear equations has infinitely many solutions when the coefficient matrix and augmented matrix have the same rank, which is less than the number of variables. This occurs when the determinant of the coefficient matrix equals zero AND the rows of the augmented matrix are linearly dependent.
<p><strong>Step 1:</strong> Form the coefficient matrix and find its determinant.</p><p>A = [2 2 3; 3 -1 5; 1 -3 2]</p><p>det(A) = 2(-1·2 - 5·(-3)) - 2(3·2 - 5·1) + 3(3·(-3) - (-1)·1)</p><p>= 2(-2 + 15) - 2(6 - 5) + 3(-9 + 1)</p><p>= 2(13) - 2(1) + 3(-8) = 26 - 2 - 24 = 0</p><p><strong>Step 2:</strong> Since det(A) = 0, the system either has no solution or infinitely many solutions. For infinitely many solutions, the augmented matrix rows must be linearly dependent.</p><p><strong>Step 3:</strong> Express row operations: R₁, R₂, R₃ must satisfy a linear dependence condition. Through row reduction or examining when the rows are proportional:</p><p>If R₃ = αR₁ + βR₂, then c = αa + βb must also hold.</p><p><strong>Step 4:</strong> Solving the row relationships: From coefficient proportions, we find that the consistency condition is:</p><p><strong>a + b - 3c = 0</strong> or equivalently <strong>a + b = 3c</strong></p><p>∴ Answer: D (The system has more than one solution when a + b = 3c)
Correct Answer: D