Determinant (Geometric Meaning) 📐

The determinant of a 2x2 matrix is a single number that reveals a surprising amount about the transformation it represents. Geometrically, the absolute value of the determinant tells you the area scaling factor of the transformation.

Imagine a unit square formed by the standard basis vectors $\mathbf{\hat{\imath}}$ and $\mathbf{\hat{\jmath}}$. This square has an area of 1. When you apply a matrix transformation, this square gets warped into a parallelogram. The area of this new parallelogram is exactly the absolute value of the determinant of the matrix.

• If $|\det(A)| = 3$, the transformation blows up areas by a factor of 3.
• If $|\det(A)| = 0.5$, it squishes areas to be half their original size.
• If $\det(A) = 0$, it squishes all of space onto a single line (or even a point), meaning the area becomes zero. This is a crucial indicator that the matrix is "singular" and does not have an inverse.

What about the sign?

The sign of the determinant tells you about the transformation's effect on orientation. If the determinant is negative, it means the transformation has flipped space, like turning it inside-out or looking at it in a mirror. In 2D, this corresponds to the order of the basis vectors being reversed.