Applications of Determinants

Volume

$|\det(A)|$ = volume of the parallelepiped formed by the columns of $A$.

• $\det > 0$: right-handed orientation
• $\det < 0$: left-handed orientation
• $\det = 0$: degenerate (flat, zero volume)

Cramer’s Rule

For $Ax = b$ with $\det(A) \neq 0$:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

Where $A_i$ is $A$ with column $i$ replaced by $b$.

Area of a Triangle

For vertices $(x_1,y_1)$, $(x_2,y_2)$, $(x_3,y_3)$:

$$\text{Area} = \frac{1}{2}\left|\det\begin{pmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{pmatrix}\right|$$