Which of the following homogeneous systems have a nontrivial solution?
5x+ y + 5z =0
−2x + 2y− 4z = 0
4x− 3y + 6z = 0
can someone please explain how you know
To determine which of the given systems of equations have a nontrivial solution, you need to check if the system is consistent or inconsistent. The system is consistent if it has at least one solution and inconsistent if it has no solution.
A homogeneous system of linear equations can be represented in matrix form as AX = 0, where A is the coefficient matrix and X is the column vector containing the variables.
For a nontrivial solution to exist, the determinant of the coefficient matrix A must be equal to zero.
Let's write the given system of equations as a matrix equation:
⎛
⎝
5 1 5
-2 2 -4
4 -3 6
⎞
⎠
⎛
⎝
x
y
z
⎞
⎠
=
⎛
⎝
0
0
0
⎞
⎠
Now, calculate the determinant of the coefficient matrix A:
det(A) = |A| = 5(2*6 - (-4)*(-3)) - 1*(-2*6 - (-4)*4) + 5*(-2*(-3) - 2*4)
Simplifying this expression, we get:
det(A) = 5(12 - 12) - (-2*6 - (-4)*4) + 5(6 + 8)
= 0 - (-12 + 16) + 5(14)
= 0 - 4 + 5(14)
= -4 + 70
= 66
Since the determinant of the coefficient matrix A is non-zero (det(A) ≠ 0), this implies that the system of equations has only the trivial solution (x = y = z = 0). Therefore, there is no nontrivial solution for this system of equations.
Hence, none of the given homogeneous systems have a nontrivial solution.