SOLUTION: Determine the values of k such that the system of linear equations does not have a unique solution. (Enter your answers as a comma-separated list.)
x + y + kz = 4
x + k
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-> SOLUTION: Determine the values of k such that the system of linear equations does not have a unique solution. (Enter your answers as a comma-separated list.)
x + y + kz = 4
x + k
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Question 1122187: Determine the values of k such that the system of linear equations does not have a unique solution. (Enter your answers as a comma-separated list.)
x + y + kz = 4
x + ky + z = 8
kx + y + z = 3
The given system of equation does not have a unique solution if and only if the determinant of the coefficient matrix is equal to zero.
The determinant is equal to .
So, the condition under the question is satisfied if and only if
= 0.
This polynomial has the roots k= 1 (of the multiplicity 2) and k= -2.
You can check on your own that the polynomial can be factored in this way
= .
Answer. The given system of linear equations does not have a unique solution if and only if k= 1 and/or k= -2.
