For the system of equations:
2x+λy=8
λx+8y=3
The system has a unique solution when the coefficient determinant is non-zero.
The coefficient matrix is:
(2λλ8)
The determinant is:
Δ=(2)(8)−(λ)(λ)
Δ=16−λ2
For a unique solution, the determinant must be non-zero:
16−λ2=0
16=λ2
λ2=16
λ=±4
Therefore, the system has a unique solution for all values of λ except λ=4 and λ=−4.
The answer is: λ=±4