Problem Statement
Consider the system of linear equations:
$$\begin{cases} x + 2y - z = 4 \\ 2x - y + 3z = 3 \\ 3x + y + kz = 7 \end{cases}$$
where $k \in \mathbb{R}$.
(a) Find the value of $k$ for which the system does not have a unique solution. [4 Marks]
(b) For this value of $k$, determine whether the system is consistent (infinitely many solutions) or inconsistent (no solutions). [3 Marks]
Verified Solution & Marking Scheme
Part (a): Evaluate the Coefficient Matrix Determinant
The system fails to possess a unique solution when $\det(A) = 0$:
$$\det(A) = \begin{vmatrix} 1 & 2 & -1 \\ 2 & -1 & 3 \\ 3 & 1 & k \end{vmatrix}$$
Expand along row 1:
$$= 1(-k - 3) - 2(2k - 9) - 1(2 - (-3))$$
$$= -k - 3 - 4k + 18 - 5 = -5k + 10$$
Setting $\det(A) = 0 \implies -5k + 10 = 0 \implies k = 2$.
Part (b): Perform Gaussian Elimination for k = 2
Augmented matrix with $k = 2$:
$$\begin{pmatrix} 1 & 2 & -1 & 4 \\ 2 & -1 & 3 & 3 \\ 3 & 1 & 2 & 7 \end{pmatrix}$$
Row operations: $R_2 \to R_2 - 2R_1$ and $R_3 \to R_3 - 3R_1$:
$$\begin{pmatrix} 1 & 2 & -1 & 4 \\ 0 & -5 & 5 & -5 \\ 0 & -5 & 5 & -5 \end{pmatrix}$$
Now $R_3 \to R_3 - R_2$:
$$\begin{pmatrix} 1 & 2 & -1 & 4 \\ 0 & -5 & 5 & -5 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
The row of zeros with a zero constant indicates that the equations are linearly dependent.
Conclusion for k = 2
Since the bottom row is $0x + 0y + 0z = 0$, the system is consistent with infinitely many solutions (representing three planes intersecting along a common line).