Evaluating Determinants

1. \(\left\vert \begin{array}{r@{\quad}r} 2 & 3 \\ -1 & 2 \end{array} \right\vert =(2)(2)-(-1)(3)=4+3=7\)
2. \[\begin{align*} \left\vert \begin{array}{r@{\quad}r@{\quad}r} A & B & C \\ 2 & 3 & 1 \\ -1 & 2 & -1 \end{array} \right\vert &=\left\vert \begin{array}{r@{\quad}r} 3 & 1 \\ 2 & -1 \end{array} \right\vert A-\left\vert \begin{array}{r@{\quad}r} 2 & 1 \\ -1 & -1 \end{array} \right\vert B+\left\vert \begin{array}{r@{\quad}r} 2 & 3 \\ -1 & 2 \end{array} \right\vert C\\ &=(-3-2)A-(-2+1)B+(4+3)C=-5A+B+7C \end{align*}\]