Matrices & Determinants
System of linear equations - consistency condition
Grade 12
Question:
<p>If the system of linear equations<br>\(x - 4y + 7z = g\)<br>\(3y - 5z = h\)<br>\(-2x + 5y - 9z = k\)<br>is consistent, then:</p>
<p>\(g + 2h + k = 0\)</p>
<p>\(g + h + 2k = 0\)</p>
<p>\(2g + h + k = 0\)</p>
<p>\(g + h + k = 0\)</p>
Step-by-Step Solution
Key Concept: A system is consistent if and only if the augmented matrix has the same rank as the coefficient matrix. Use row reduction to find the constraint relating g, h, and k that makes the system solvable.
<p><strong>Step 1:</strong> Write the augmented matrix and perform row reduction on the coefficient matrix.</p><p>Coefficient matrix:</p><p>$$\begin{pmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{pmatrix}$$</p><p><strong>Step 2:</strong> Apply R₃ → R₃ + 2R₁:</p><p>$$\begin{pmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ 0 & -3 & 5 \end{pmatrix}$$</p><p><strong>Step 3:</strong> Apply R₃ → R₃ + R₂:</p><p>$$\begin{pmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ 0 & 0 & 0 \end{pmatrix}$$</p><p><strong>Step 4:</strong> The third row becomes [0 0 0] in the coefficient matrix. For consistency, the corresponding augmented entry must also be zero.</p><p>Applying the same operations to the RHS: k + 2g + h must equal 0</p><p><strong>Step 5:</strong> The consistency condition is:</p><p>$$k + 2g + h = 0$$ or equivalently $$2g + h + k = 0$$</p><p>∴ Answer: A (the relation is 2g + h + k = 0)</p>
Correct Answer: A