Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 18 / b

The characteristic polynomial of $A$ isIts distinct eigenvalues are $3$ and $-3$. For $3$, the eigenvector relation gives $y=-2x$, so choose $(1,-2{)}^T$. For $-3$, it gives $x=-2y$, so choose $(2,-1{)}^T$. Put these independent eigenvectors into columns:Then $AB=BC$ column by column. Also $\det B=3\ne 0$, so this choice supplies the diagonalization of a matrix $B^{-1}AB=C$. Scaling columns or interchanging their order gives other valid choices, with the matching order in $C$.

For the choice of $B$ in part (b)(i), the two-by-two matrix inverse formula givesMultiplication gives $B{B}^{-1}=B^{-1}B=I$, providing a direct check.