Matrices & Determinants
System of linear equations - infinite solutions
Grade 12
Question:
<p>If 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, has more than one solution, then:</p>
<p>\(b - c - a = 0\)</p>
<p>\(a + b + c = 0\)</p>
<p>\(b - c + a = 0\)</p>
<p>\(b + c - a = 0\)</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 unknowns. This occurs when the rows of the augmented matrix are linearly dependent, imposing a constraint relating a, b, and c.
<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 has either no solution or infinitely many solutions. For infinitely many solutions, the augmented matrix must have rank equal to rank(A).</p><p><strong>Step 3:</strong> Apply row operations on augmented matrix [A|b]:</p><p>R₂ → R₂ - (3/2)R₁, R₃ → R₃ - (1/2)R₁</p><p>After simplification, the condition for consistency (rank of augmented matrix = rank of A) yields:</p><p><strong>a + b - c = 0</strong> or equivalently <strong>a + b = c</strong></p><p>∴ Answer: D</p>
Correct Answer: D