Homogenous system - Nontrivial solutions

[shiri]

For which values of 'a' does the homogeneous system

ax1 + x2 - x3 = 0
x1 - x2 + 2x3 = 0
2x1 + 3x2 + x3 = 0

have a nontrivial solution? Find all the nontrivial solutions.

 

[berkeman]

For which values of 'a' does the homogeneous system

ax1 + x2 - x3 = 0
x1 - x2 + 2x3 = 0
2x1 + 3x2 + x3 = 0

have a nontrivial solution? Find all the nontrivial solutions.

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[Architeuthis Dux]

I would like to clarify that a homogeneous system refers to a system of linear equations in which all the constants on the right-hand side of the equations are equal to zero. In simpler terms, it is a system of equations with no constant term.

To determine the values of 'a' for which the given homogeneous system has a nontrivial solution, we can use the concept of determinant. The determinant of a matrix is a value that can be calculated from the elements of a square matrix and is used to solve systems of linear equations.

In this case, we can write the system of equations in matrix form as follows:

|a 1 -1| |x1| |0|
|1 -1 2| |x2| = |0|
|2 3 1| |x3| |0|

The determinant of this matrix is given by:
det = a(1)(1) + (-1)(2)(3) + (-1)(2)(1) - (-1)(1)(2) - (1)(3)(a) - (1)(-1)(1)
det = a - 6 + 2 - 2a + 3a + 1
det = 2a - 5

For a nontrivial solution to exist, the determinant must be equal to zero. Therefore, we can set 2a - 5 = 0 and solve for 'a':
2a - 5 = 0
2a = 5
a = 5/2

Hence, for the given homogeneous system to have a nontrivial solution, the value of 'a' must be equal to 5/2.

To find all the nontrivial solutions, we can use the method of Gaussian Elimination or any other method of solving systems of linear equations. However, since the system is already in reduced echelon form, we can simply write the solutions as:

x1 = 5/4, x2 = 1/4, x3 = -1/2

Therefore, the nontrivial solution to the given homogeneous system is (5/4, 1/4, -1/2).

In conclusion, a homogeneous system with nontrivial solutions has a unique value of 'a', which in this case is 5/2. The nontrivial solution can be found by setting the determinant