The question is asking for what values of x will the matrix have at least one repeated eigenvalue (algebraic multiplicity of 2 or greater). The matrix is | 3 0 0 | | 0 x 2 | So naturally a normal attempt to find the eigenvalue in a question with only intergers | 0 2 x | I would continue with: | [tex]\lambda[/tex]-3 0 0 | | 0 [tex]\lambda[/tex]-x -2| | 0 -2 [tex]\lambda[/tex]-x| And then finding the determinant would continue with ([tex]\lambda[/tex]-3)([tex]\lambda[/tex]-x)([tex]\lambda[/tex]-x) - (-2)(-2)([tex]\lambda[/tex]-3) etc... Except with 2 unknown variable it inevitably becomes a problem that I run into with more theoretical questions vs questions dealing only with numbers. As far as I know an eigenvalue with an algebraic multiplicity of >1 doesn't even have a geometric significance. Clearly I am approaching this entire question from the wrong angle and could use a push in the right direction if anyone is so able. Muchly appreciated.