Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 6 / i / Solution

For an idempotent matrix, $A^k=A$ for every integer $k\ge 1$. Thus any scalar polynomial reduces toor equivalently $aI+bA$. This expression describes all polynomials in $A$, although its two coefficients need not be unique if $A=0$ or $A=I$.