Answer to Question #235665 in Linear Algebra for Amuj

Question #235665
For which rational numbers does the homogenous system
x+(λ-3)y=0
(λ-3)x+y=0
Have a non-trivial solution
1
Expert's answer
2021-09-15T03:42:15-0400

"\\text{The co-efficient matrix for the given system is}\n\\begin{pmatrix}\n1 & \\lambda - 3 \\\\\n\\lambda -3 & 1\n\\end{pmatrix}\\\\\n\\text{For given system of linear equations to have a non-trivial solution, then its determinant}\\\\\n\\text{be 0.}\\\\\n\\begin{vmatrix}\n1 & \\lambda - 3 \\\\\n\\lambda -3 & 1\n\\end{vmatrix}= 1-(\\lambda -3)^2 =0\\\\\n= -\\lambda^2 + 6\\lambda-8=0\\\\\n\\implies \\text{$\\lambda = -2$ or $\\lambda=-4$}"


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