# How do you find the determinant of |(3,4), (2,5)|?

Jan 18, 2017

$\det \left(\begin{matrix}3 & 4 \\ 2 & 5\end{matrix}\right) = 7$

#### Explanation:

The use of proper notation should be observed, as:

$| \left(3 , 4\right) , \left(2 , 5\right) | \text{ }$, or $\text{ } \det \left(\begin{matrix}3 & 4 \\ 2 & 5\end{matrix}\right)$

is the determinant, and you cannot find the determinant of a determinant. The question should be, find the determinant of

$\left(\begin{matrix}3 & 4 \\ 2 & 5\end{matrix}\right)$

Which is a 2x2 matrix. The determinant of a 2x2 matrix can be calculated as follows:

$\det \left(\begin{matrix}a & b \\ c & d\end{matrix}\right) = | \left(a , b\right) , \left(c , d\right) | = a d - b c$

Hence:

$\det \left(\begin{matrix}3 & 4 \\ 2 & 5\end{matrix}\right) = | \left(3 , 4\right) , \left(2 , 5\right) |$
$\text{ } = \left(3\right) \left(5\right) - \left(2\right) \left(4\right)$
$\text{ } = 15 - 8$
$\text{ } = 7$