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 x 2 determinants and area, and a link between 3 x 3 determinants and volume.

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

a x b = $\displaystyle \left\vert\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array} }\right.$ $\displaystyle \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} a_1 & a_2 & a_3 b_1 & b_2 & b_3 \end{array} }\right\vert$ .

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 x b = $\displaystyle \left\vert\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right.$ $\displaystyle \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right\vert$ k.

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

$\displaystyle \left\vert\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right.$ $\displaystyle \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right\vert$ .

||a x b|| = $\displaystyle \left\vert\vphantom{\det \left(\left[ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} \right]\right) }\right.$ det $\displaystyle \left(\vphantom{\left[ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} \right]}\right.$ $\displaystyle \left[\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right.$ $\displaystyle \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} }\right]$ $\displaystyle \left.\vphantom{\left[ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} \right]}\right)$ $\displaystyle \left.\vphantom{\det \left(\left[ \begin{array}{cc} a_1 & a_2 b_1 & b_2 \end{array} \right]\right) }\right\vert$ .

The volume of a parallelepiped spanned by the vectors a, b and c is the absolute value of the scalar triple product (a x 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 x b)^.c = $\displaystyle \left\vert\vphantom{ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} }\right.$ $\displaystyle \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} }\right\vert$ .

|(a x b)^.c| = $\displaystyle \left\vert\vphantom{\det \left(\left[ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} \right]\right) }\right.$ det $\displaystyle \left(\vphantom{\left[ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} \right]}\right.$ $\displaystyle \left[\vphantom{ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} }\right.$ $\displaystyle \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} }\right]$ $\displaystyle \left.\vphantom{\left[ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} \right]}\right)$ $\displaystyle \left.\vphantom{\det \left(\left[ \begin{array}{ccc} a_1 & a_2 & a_3 b_1 & b_2 & b_3 c_1 & c_2 & c_3 \end{array} \right]\right) }\right\vert$ .