Answer to Question #339479 in Linear Algebra for Mmq

Question #339479

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

1
Expert's answer
2022-05-13T10:49:59-0400

An "n\u00d7n" 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}\n 1 & 1 & \\alpha \\\\\n 1 & 1 & \\beta \\\\\n \\alpha & \\beta & 1 \\\\\n\\end{vmatrix}=1\\begin{vmatrix}\n 1 & \\beta \\\\\n \\beta & 1\n\\end{vmatrix}-1\\begin{vmatrix}\n 1 & \\beta \\\\\n \\alpha & 1\n\\end{vmatrix}+\\alpha\\begin{vmatrix}\n 1 & 1 \\\\\n \\alpha & \\beta\n\\end{vmatrix}"

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

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

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


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