Determine whether or not each of the following homogeneous systems has a nonzero solution:
a. x + y + z = 0
2x − 3y + z = 0
x − 4y + 2z = 0
b. x + y + z = 0
2x + 4y − z = 0
3x + 2y + 2z = 0
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To determine if a homogeneous system has a nonzero solution, we need to find the determinant of the coefficient matrix. If the determinant is not equal to zero, then the system has a nonzero solution. Otherwise, it does not.
a. The coefficient matrix is:
1 1 1
2 -3 1
1 -4 2
Calculating the determinant, we get:
det = 1(-3)(2) + 1(1)(1) + 1(2)(-4) - 1(-3)(1) - 2(1)(2) - 1(1)(-4)
= -6 + 1 - 8 + 3 - 4 - 4
= -18
Since the determinant is not equal to zero (-18 ≠ 0), the system has a nonzero solution.
b. The coefficient matrix is:
1 1 1
2 4 -1
3 2 2
Calculating the determinant, we get:
det = 1(4)(2) + 1(1)(3) + 1(2)(2) - 1(4)(3) - 2(1)(2) - 1(2)(2)
= 8 + 3 + 4 - 12 - 4 - 4
= -5
Since the determinant is not equal to zero (-5 ≠ 0), the system has a nonzero solution.
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