Matrices & Determinants
System of linear equations with infinite solutions
Grade 12
Question:
<p>Given the system of linear equations:<br>\(x - 2y + kz = 1\) (1)<br>\(2x + y + z = 2\) (2)<br>\(3x - y - kz = 3\) (3)<br>For the system to have infinite solutions, the value of <em>k</em> is:</p>
<p>\(k = \dfrac{1}{2}\)</p>
<p>\(k = -\dfrac{1}{2}\)</p>
<p>\(k = 2\)</p>
<p>\(k = -2\)</p>
Step-by-Step Solution
Key Concept: For infinite solutions, the system must be consistent and dependent, meaning the coefficient matrix and augmented matrix must have equal rank < 3. This requires the determinant of the coefficient matrix to be zero AND the augmented matrix rows to be linearly dependent.
<p><strong>Step 1:</strong> Write the coefficient matrix A and compute its determinant.</p><p>$$A = \begin{bmatrix} 1 & -2 & k \\ 2 & 1 & 1 \\ 3 & -1 & -k \end{bmatrix}$$</p><p><strong>Step 2:</strong> Calculate det(A) by expanding along row 1:</p><p>$$\det(A) = 1(1·(-k) - 1·(-1)) - (-2)(2·(-k) - 1·3) + k(2·(-1) - 1·3)$$</p><p>$$= 1(-k + 1) + 2(-2k - 3) + k(-2 - 3)$$</p><p>$$= -k + 1 - 4k - 6 - 5k = -10k - 5$$</p><p><strong>Step 3:</strong> For infinite solutions, set det(A) = 0:</p><p>$$-10k - 5 = 0 \implies k = -\frac{1}{2}$$</p><p><strong>Step 4:</strong> Verify consistency by row reduction of augmented matrix [A|B] with k = -1/2. Check that equations are dependent (one equation is a linear combination of others) and no contradiction arises. Substitution confirms the system reduces to 2 independent equations in 3 unknowns.</p><p>∴ <strong>Answer: B</strong> (k = -1/2)</p>
Correct Answer: B