Unit content
Determinants
A square matrix represents a linear transformation of a space into itself. The determinant summarizes how that transformation scales oriented area or volume.
For a square matrix $A$, its determinant is written
$$\det A$$
or $|A|$.
Scaling area and volume
In two dimensions, if a transformation represented by $A$ sends a region of area $S$ to a region of area $S'$, then
$$S'=|\det A|S.$$
In three dimensions, the same absolute value gives the volume-scaling factor.
For example, a transformation that doubles every coordinate in $\mathbb R^2$ has matrix
$$A=\begin{pmatrix}2&0\0&2\end{pmatrix}$$
and determinant $4$, so areas are multiplied by $4$.
Orientation
The sign of the determinant records orientation. A positive determinant preserves orientation, while a negative determinant reverses it, as a reflection does.
Zero determinant
If
$$\det A=0,$$
some dimension is collapsed. A plane may be flattened onto a line, for example. Distinct input vectors can then produce the same output, so the transformation cannot be inverted.
For a square matrix,
$$A\text{ is invertible}\qquad\Longleftrightarrow\qquad\det A\ne0.$$
The determinant therefore combines geometric scaling, orientation and invertibility into a single scalar.