Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
The system of equations $6x + 5y + \lambda z = 0, 3x - y + 4z = 0, x + 2y - 3z = 0$ has:
Only a trivial solution for $\lambda \in \mathbb{R}$
Exactly one non-trivial solution for some real $\lambda$
Infinite number of non-trivial solutions for one value of $\lambda$
Only one solution for $\lambda \neq -5$
Step-by-Step Solution
Key Concept: A homogeneous system of linear equations has non-trivial solutions if and only if the determinant of the coefficient matrix equals zero. For this system, det(A) = 0 yields λ = -5, and when λ = -5, the rank of the coefficient matrix becomes less than 3, allowing infinite non-trivial solutions.
For the system to have a non-trivial solution, the determinant of the coefficient matrix must equal zero. Computing $\begin{vmatrix} 6 & 5 & \lambda \\ 3 & -1 & 4 \\ 1 & 2 & -3 \end{vmatrix} = 0$ yields $\lambda = -5$. When $\lambda = -5$, the only solution is $x = 0, y = 0, z = 0$, confirming that a non-trivial solution does not exist for any value of $\lambda$.
Correct Answer: 3,4