An elementary use that was in the books on determinants (not linear algebra) I read as a student long ago: you can use them to write cute equations for objects in elementary (Euclidean or projective) geometry. For instance, the equation for the circle through three points in the plane is $$ \left| \begin{matrix} x^2+y^2 & x & y & 1 \cr a_1^2+a_2^2 & a_1 & a_2 & 1\cr b_1^2+b_2^2 & b_1 & b_2 & 1\cr c_1^2+c_2^2 & c_1 & c_2 & 1 \end{matrix} \right| = 0 $$ If the coefficient of $x^2+y^2$ is non zero then it is obviously the equation for a circle, and the three points $A, B, C$ obviously lie on the circle.
The coefficient of $x^2+y^2$ is $$ \left| \begin{matrix} a_1 & a_2 & 1\cr b_1 & b_2 & 1\cr c_1 & c_2 & 1 \end{matrix} \right| $$ and has to be nonzero. Since the equation for the line through $(b_1, b_2)$ and $(c_1, c_2)$ is $$ \left| \begin{matrix} x_1 & x_2 & 1\cr b_1 & b_2 & 1\cr c_1 & c_2 & 1 \end{matrix} \right|=0 $$ the coefficient of $x^2+y^2$ is nonzero if the three points do not lie on a straight line.

In the same vein, the linear homogeneous differential equation satisfied by $y_1(x)$ and $y_2(x)$ is $$ \left| \begin{matrix} y''(x) & y'(x) & y(x) \cr y_1''(x) & y_1'(x) & y_1(x) \cr y_2''(x) & y_2'(x) & y_2(x) \end{matrix} \right| = 0 $$