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If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z = 0 has a non-zero solution, then the possible values of k are
- a)- 1, 2
- b)1, 2
- c)0, 1
- d)- 1, 1
Correct answer is option 'D'. Can you explain this answer?
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If the system of equations x - ky - z = 0, kx - y - z = 0, x + y - z =...
Solution:
Given system of equations:
x - ky - z = 0
kx - y - z = 0
x + y - z = 0
Let A be the coefficient matrix of the system of equations, i.e.,
A = [1 -k -1]
[k -1 -1]
[1 1 -1]
Let B = [x]
[y]
[z]
Then, the system of equations can be written as AB = 0.
For non-zero solution, the determinant of A must be zero, i.e., det(A) = 0.
Hence, we have:
det(A) = 1(-1)(-1) + k(-1)(-1) + (1)(k)(1) - (1)(-1)(1) - k(1)(-1) - (1)(1)(-1) = 0
Simplifying the above equation, we get:
k² - 2k - 2 = 0
Solving the above quadratic equation, we get:
k = (2 ± √8)/2 = 1 ± √2
Since k must be real, we have k = 1 - √2 or k = 1 + √2.
Therefore, the possible values of k are -1, 1 + √2 and 1 - √2.
But, we know that the system of equations has non-zero solution only if k = 1, as for k = -1, the system of equations become inconsistent and for k = 1 + √2 and k = 1 - √2, the system of equations become dependent.
Hence, the only possible value of k is 1.
Therefore, the correct option is (D).
Given system of equations:
x - ky - z = 0
kx - y - z = 0
x + y - z = 0
Let A be the coefficient matrix of the system of equations, i.e.,
A = [1 -k -1]
[k -1 -1]
[1 1 -1]
Let B = [x]
[y]
[z]
Then, the system of equations can be written as AB = 0.
For non-zero solution, the determinant of A must be zero, i.e., det(A) = 0.
Hence, we have:
det(A) = 1(-1)(-1) + k(-1)(-1) + (1)(k)(1) - (1)(-1)(1) - k(1)(-1) - (1)(1)(-1) = 0
Simplifying the above equation, we get:
k² - 2k - 2 = 0
Solving the above quadratic equation, we get:
k = (2 ± √8)/2 = 1 ± √2
Since k must be real, we have k = 1 - √2 or k = 1 + √2.
Therefore, the possible values of k are -1, 1 + √2 and 1 - √2.
But, we know that the system of equations has non-zero solution only if k = 1, as for k = -1, the system of equations become inconsistent and for k = 1 + √2 and k = 1 - √2, the system of equations become dependent.
Hence, the only possible value of k is 1.
Therefore, the correct option is (D).
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