<s>[INST] What are all the uses of the determinant? I've learned how to calculate the determinant but what is the determinant used for? So far, I only know that there is no inverse if the determinant is 0.
 [/INST] *

*It allows you to evaluate cross products and find the general equation of the plane if given $3$ points. For instance: $A(1,1,0),\, B =(1,0,1),\,C=(0,1,2)$  $$B-A=(1,0,1)-(1,1,0)=(0,-1,1)$$ and $$C-A=(0,1,2)-(1,1,0)=(-1,0,2)$$ You now use the cross-product of $$(B-A)\times(C-A)=\begin{bmatrix}i & j & k \\0 & -1 & 1 \\-1 & 0 & 2 \end{bmatrix}=(-2,-1,-1)= \vec{n}$$ $i$, $j$ and $k$ are unit vectors that are directed along the $x$, $y$ and $z$ axis respectively. $\vec{n}$ is known as the normal vector $\vec{n}$ and is perpendicular to the equation of the plane.


*For square matrices formulae such as $$\det(AB) = \det (A) \det (B)$$ and $$\det (A^{-1}) = \frac 1 {\det (A)}$$ are applicable

*Allows you to check if your matrix has an inverse: $$\det=\begin{vmatrix}0 & 0 & 2 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & -1 & 3 & 0\\ 2 & 1 & 5 & -3 \end{vmatrix}$$  
$$=0\times\begin{vmatrix} 0 & 0 & 1 \\  -1 & 3 & 0 \\ 1 & 5 & -3 \end{vmatrix}+0\times\begin{vmatrix} 1 & 0 & 1 \\  0 & 3 & 0 \\ 2 & 5 & -3 \end{vmatrix}+2\times\begin{vmatrix} 1 & 0 & 1 \\  0 & -1 & 0 \\ 2 & 1 & -3 \end{vmatrix}+0\times\begin{vmatrix} 1 & 0 & 0 \\  0 & -1 & 3 \\ 2 & 1 & 5 \end{vmatrix}$$$$=2\times\begin{vmatrix} 1 & 0 & 1 \\  0 & -1 & 0 \\ 2 & 1 & -3 \end{vmatrix}$$ $$=2\left( 1\times\begin{vmatrix}-1 & 0 \\ 1 & -3 \\ \end{vmatrix}+0\times\begin{vmatrix}0 & 0 \\ 2 & -3 \\ \end{vmatrix}+ 1\times\begin{vmatrix}0 & -1 \\ 2 & 1 \\ \end{vmatrix}\right)$$$$=2\times \left(1\times(3-0)+1\times(0--2)\right)= 2(3+2)=10 \ne 0$$ So your matrix is invertible (or has an inverse).
</s>