Chapter 5 - Section 5.3 - Determinant Solution of Linear Systems - 5.3 Exercises - Page 512: 7

$-26$

The determinant for a $2 \times 2$ matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}$ can be defined as: $\text{det}= \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$ Therefore, $\text{det}=\begin{vmatrix} 3 & 4 \\ 5 & -2 \\ \end{vmatrix}=(3)(-2) -(4)(5) =-6-20 =-26$ Therefore, the determinant of the given matrix is $-26$.