Determinant

The determinant can only be calculated from matrices with equal number of rows and columns. The determinant is a single value meaning the scale change of the space.

Image a 2-dimensional euclidean space $\mathbb{R}^2$ space, the unit vector on $\mathbf{x}$ and $\mathbf{y}$ constructs a small square. After a linear transformation into a new 2-d space, how much does the space of the square changed? If it grows to 2 times the original space, the determinant of the transformation matrix would be 2. If the determinant is negative, it means the space is flipped.