Precalc Scalars and Matrices

Actually

$\mathbf{P} = \begin{pmatrix} 2 & -1 \\ 3 & 1 \end{pmatrix}$

You have   $\displaystyle \textbf{v}=\left[\begin{array}{c} 2a+3b\\ -a+b \end{array}\right]$ and require that  $\displaystyle P\, \textbf{v}=\left[\begin{array}{c} 2a \\ -a \end{array} \right]\, .$

So, let  $\displaystyle P=\left[\begin{array}{cc} r & s \\ t & u \end{array}\right]$ say, multiply out the rhs and equate components across the equation to find r, s, t and u.

You should find that $\displaystyle P = \left[ \begin{array}{cc} 2/5 & -6/5 \\ -1/5 & 3/5 \end{array} \right]\, .$