Determinants measure how space stretches or compresses. See area and volume scaling come alive.
The determinant of a matrix is a single number that captures something profound: how much the transformation scales area (in 2D) or volume (in 3D).
But it tells us even more. The sign reveals whether orientation is preserved or flipped. And when the determinant is zero? That's when space collapses—and inverses cease to exist.
For a 2×2 matrix: det(A) = ad - bc
The unit square has area 1. After transformation, it becomes a parallelogram whose area equals the absolute value of the determinant.
The determinant equals the factor by which area scales. The unit square (area = 1) becomes a parallelogram with area = |det|.
A reflection flips orientation—counterclockwise becomes clockwise. This is indicated by a negative determinant.
The sign of the determinant tells you about orientation. Positive = preserved (counterclockwise stays counterclockwise). Negative = flipped (like looking in a mirror).
In 3D, the determinant of a 3×3 matrix tells us how volume scales. The unit cube becomes a parallelepiped.
In 3D, the determinant equals the factor by which volume scales. The unit cube (volume = 1) becomes a parallelepiped with volume = |det|.
When det = 0, the transformation collapses space to a lower dimension. A 2D region might become a line or even a single point.
When the determinant is zero, space collapses. A 2D region can become a line (rank 1) or even a single point (rank 0). This loss of dimension is why no inverse exists.
Next: Eigenvectors—the special directions that transformations only stretch, never rotate.