Determinants: area & volume
A determinant turns a square matrix into a number. Geometrically, it tells us how much signed area or volume the column vectors enclose. Let us first find out why its 2D formula is ad − bc.
01Why ad − bc? Watch the area stay put.
Start both vectors at the origin. Complete their opposite sides to enclose a parallelogram. Read the matrix down its columns: u = (a, c), v = (b, d). Dashed component lines show exactly which horizontal and vertical displacement each letter means.
Original matrix and columns
Here a, b, c and d are positive, and ad > bc. We start with this simple case so each area can be seen directly.
Both rearrangements are shears: they slide parallel strips without stretching those strips. That is why we can measure the final rectangle instead of the original parallelogram. The small strokes merely make the sliding visible; the map applies to the whole continuous region.
We use columns to match the matrix-transformation lesson. You could instead use the rows (a, b) and (c, d): transposing a matrix leaves its determinant unchanged, so the enclosed area is the same.
02Zero area means one direction was lost.
The determinant is a signed area. Its absolute value is the geometric area; its sign tells the orientation of the ordered pair u, v. Now bring v onto the same line as u. The parallelogram becomes thinner until it has no area.
This is linear dependence: one vector can be made by scaling the other. A zero vector also creates dependence. For a nonzero determinant, turning from u to v through the smaller angle is counterclockwise for a positive sign and clockwise for a negative sign. A negative determinant does not mean negative geometric area.
03In three dimensions, area becomes volume.
Use three vectors from the same origin. Their sums give the eight corners of a parallelepiped: a slanted box. These three vectors are the columns of a 3 × 3 matrix. The absolute value of its determinant is the volume of that box. Drag the model to look around it.
u · column 1 v · column 2 w · column 3
Initially the base lies in the xy-plane: base area = 2 × 1.7 = 3.4, perpendicular height = 1.8, and volume = 3.4 × 1.8 = 6.12. A sloping edge is not the perpendicular height.