Question

By examining the determinant of the coefficient matrix, show that the following system has a nontrivial solution if and only if α = β

x + y + αz = 0

x + y + βz = 0

αx + βy + z = 0

Answer 1

An $n×n$ homogeneous system of linear equations has an unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. i.e. for a non-trivial solution $D=0.$

$$D=\begin{vmatrix} 1 & 1 & \alpha \\ 1 & 1 & \beta \\ \alpha & \beta & 1 \\ \end{vmatrix}=1\begin{vmatrix} 1 & \beta \\ \beta & 1 \end{vmatrix}-1\begin{vmatrix} 1 & \beta \\ \alpha & 1 \end{vmatrix}+\alpha\begin{vmatrix} 1 & 1 \\ \alpha & \beta \end{vmatrix}$$

$$=1-\beta^2-1+\alpha\beta+\alpha\beta-\alpha^2$$

$$=-(\alpha-\beta)^2$$

$$D=0=>-(\alpha-\beta)^2=0=>\alpha=\beta$$