The homogeneous system of equations given by ax1 + x2 - x3 = 0, x1 - x2 + 2x3 = 0, and 2x1 + 3x2 + x3 = 0 has nontrivial solutions when the determinant of the coefficient matrix is zero. The determinant is calculated as a function of 'a', leading to the condition that a must equal -1 or 3 for nontrivial solutions to exist. The nontrivial solutions can be expressed in terms of free variables derived from the row-reduced echelon form of the system.
PREREQUISITESStudents of linear algebra, educators teaching systems of equations, and anyone seeking to deepen their understanding of homogeneous systems and their solutions.
shiri said: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.