Significance of determinant of $n \times n$ Matrix?

Question

In $2 \times 2$ matrix, if we consider every column as a $2$-D vector, then determinant gives us cross product. But what happen when we deal with $n \times n$ matrix?

Answer 1

In general, the determinant of a matrix (linear map) $M\colon\mathbf R^n\to\mathbf R^n$ tells you the $n$-dimensional volume scaling factor.

In the $2\times 2$ case ($n=2$), any region $R$ in the plane having area $A$ will have area $A\det(M)$ after being transformed by $M$.

So for example, a 2D shape with area $5$ in the plane will have area $25$ if it is transformed by $\left(\begin{smallmatrix}3&1\\1&2\end{smallmatrix}\right)$. You can test this with images using Mathematica:

Analogously, in the $3\times 3$ case ($n=3$), any region $R$ in space having volume $V$ will have volume $V\det(M)$ after being transformed by $M$.

This generalises nicely for arbitrary $n\in\mathbf N$ when one considers "hypervolume".

Answer 2

In 2D, the determinant is the area of the parallelogram built on the two vectors.

In particular, if the two vectors are parallel, the area is null. If the vectors are orthogonal, the area is the product of the lengths.

In 3D, it is the volume of the parallelepiped build on the three vectors.

In particular, if the three vectors are coplanar, the volume is null. If the vectors are orthogonal, the volume is the product of the lengths.

And so on.