The relationship between determinants and area or volume

From the properties of the cross product and the scalar triple product, we can discover a link between 2 × 2 determinants and area, and a link between 3 × 3 determinants and volume.

2 × 2 determinants and area

Recall that the area of the parallelogram spanned by a and b is the magnitude of a×b. We can write the cross product of a = a1i + a2j + a3k and b = b1i + b2j + b3k as the determinant

a × b = | | | i j k | ||a a a || || 1 2 3|| b1 b2 b3.

Now, imagine that a and b lie in the plane so that a3 = b3 = 0. Using our rules for calculating determinants, we see that, in this case, the cross product simplifies to

a × b = || || | a1 a2 | | b1 b2 |k.

Hence, the area of the parallelogram, ||a × b||, is the absolute value of the determinant

| | ||a1 a2 || |b1 b2 |.
As mentioned in our discussion about determinant notation, it’s difficult to represent the absolute value of a determinant using the above notation. Instead, we write that the area of the parallelogram spanned by a = a1i + a2j and b = b1i + b2j is
||a × b|| = | ([ ])| || a1 a2 || |det b1 b2 |.

3 × 3 determinants and volume

The volume of a parallelepiped spanned by the vectors a, b and c is the absolute value of the scalar triple product (a × b) ⋅ c. We can write the scalar triple product of a = a1i + a2j + a3k, b = b1i + b2j + b3k, and c = c1i + c2j + c3k as the determinant

(a × b) ⋅ c = || || | a1 a2 a3 | || b1 b2 b3 || | c1 c2 c3 |.
Hence, the volume of the parallelepiped spanned by a, b, and c is
|(a × b) ⋅ c| = || ( ⌊ a a a ⌋ ) || || ( ⌈ 1 2 3 ⌉ ) || |det b1 b2 b3 | | c1 c2 c3 |.