Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 7 / Solution

For a diagonal matrix, each diagonal entry separately obeys $d^2+2d-15=(d-3)(d+5)=0$. Hence all four possibilities areFor the second form, $X^2=I$ givesBecause $I,X$ are linearly independent and $b\ne 0$, the $X$ coefficient forces $a=-1$; the $I$ coefficient then gives $b^2=16$. ThusBoth constants are nonzero, as required.