What is the geometric interpretation of the determinant of a matrix?

Here's the geometric intuition: **In 2D:** For matrix $A = [\mathbf{a}_1 \; \mathbf{a}_2]$, $|\det(A)|$ is the area of the parallelogram spanned by $\mathbf{a}_1$ and $\mathbf{a}_2$. If $\mathbf{a}_1$ and $\mathbf{a}_2$ are at an angle $\theta$, the area is $\|\mathbf{a}_1\| \|\mathbf{a}_2\| \sin \theta$, which equals $|\det(A)|$. **In 3D:** The absolute determinant is the volume of the parallelepiped. **The sign indicates orientation:** - Positive: the transformation preserves orientation (right-hand rule) - Negative: the transformation flips orientation (mirror image) **Why zero determinant means singular:** If $\mathbf{a}_1$ and $\mathbf{a}_2$ are linearly dependent (one is a scalar multiple of the other), they span zero area — the parallelogram collapses. This means the transformation crushes the space into a lower dimension, so it's not invertible. **Scaling factor:** If you apply a linear transformation $A$ to a region of volume $V$, the new volume is $|\det(A)| \cdot V$. This is why the determinant appears in the change of variables formula for multiple integrals.