Chapter 12 - Matrices - 12-3 Determinants and Inverses - Practice and Problem-Solving Exercises - Page 788: 17

$11$

The determinant of a $2\times2$ matrix $A=\begin{bmatrix}a & b\\c&d \end{bmatrix}$ is given by $ad-bc.$ Hence, the determinant of the given matrix, $ \begin{bmatrix} 5&3\\-2&1 \end{bmatrix}$ is \begin{align*}\require{cancel} & 5(1)-(-2)(3) \\&= 5-(-6) \\&= 5+6 \\&= 11 .\end{align*}