2 × 2 and 3 × 3 determinants

The determinant is a number attached to a square matrix. Its algebraic definition looks arbitrary; its geometric meaning is crystal clear: it is the factor by which areas are multiplied.

The 2 × 2 determinant

        [ a  b ]
  A =   [ c  d ]        det(A) = ad - bc

Example: det [[3, 1], [5, 2]] = 3×2 - 1×5 = 1.

What that number measures

The two columns of A are two vectors of the plane. They span a parallelogram, and the determinant gives its area — up to sign.

          ^
          |        ____________
        d |       /           /
          |      /   area =  /
          |     /  |ad - bc| /
          |    /___________/
          +-------------------->
                   a

   column 1 = (a ; c)      column 2 = (b ; d)

In the canonical basis the unit square has area 1, and its image under A has area |det A|. Hence the dynamic reading: |det A| is the area magnification factor of the associated linear map.

det = 2     ->  areas double
det = 1     ->  areas are preserved (rotation, shear)
det = 0.5   ->  areas are halved
det = 0     ->  everything is flattened onto a line or a point

The case det = 0

This is the most important one. det(A) = 0 means the parallelogram is flat: the two columns are collinear, they point in the same direction.

    det ≠ 0                      det = 0
      ^                            ^
      |  /                         |   /
      | /                          |  /  (both vectors
      |/____>                      |_/____>   on the same line)

   genuine area                  zero area
   invertible matrix             non-invertible matrix

Everything seen so far comes together here: collinear columns = dependent family = insufficient rank = system without a unique solution = non-invertible matrix.

The sign: a matter of orientation

The determinant is signed. Its sign says whether the transformation preserves the orientation of the plane or reverses it:

det > 0   ->  orientation preserved   (rotation, magnification)
det < 0   ->  orientation reversed    (reflection, mirror effect)

Reflection in the x-axis, with matrix [[1, 0], [0, -1]], has determinant -1: it preserves areas but flips the plane.

In dimension 3: a volume

For a 3 × 3 matrix, the determinant gives the volume of the parallelepiped built on the three columns. It can be computed by the rule of Sarrus:

  | a  b  c |
  | d  e  f |  = aei + bfg + cdh - ceg - afh - bdi
  | g  h  i |

  Copy the first two columns to the right,
  add the three descending diagonals,
  subtract the three ascending ones.

Careful: Sarrus only works in dimension 3. Applying it to a 4 × 4 matrix is a classic and wrong move.

Summary

• det [[a, b], [c, d]] = ad - bc.
• |det A| is the area of the parallelogram of the columns — the volume in dimension 3.
• It is the factor by which the linear map multiplies areas.
• det A = 0 ⟺ collinear columns ⟺ non-invertible matrix.
• The sign tells whether orientation is preserved (> 0) or reversed (< 0).
• Sarrus applies in dimension 3 only.