Chapter 5 - Section 5.3 - Determinant Solution of Linear Systems - 5.3 Exercises - Page 512: 1

$-31$

The determinant for a $2 \times 2$ matrix $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}$ can be defined as: $$\text{det}= \begin{vmatrix} a_{11} & a_{12} \\4 a_{21} & a_{22} \\ \end{vmatrix}=a_{11}a_{22}-a_{12}a_{21}$$ Therefore, $\text{det}=\begin{vmatrix} -5 & 9 \\ 4 & -1 \\ \end{vmatrix}=(-5)(-1) -(9)(4) =5-36 =-31$ Therefore, the determinant of the given matrix is: $-31$.