Proof That the Vector Triple Product for the Columns of a Matrix Equals the Vector Triple Product for the Rows
The scalar triple product of the columns of a three by three matrix is equal to the scalar triple product of the rows.
Proof
Take a matrix
\[ \left( \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right) \]
with columns \[\mathbf{a} , \: \mathbf{b} , \: \mathbf{c}.\]
. The scalar triple product of the columns is\[\begin{equation} \begin{aligned} \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) &= \begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix} \cdot \begin{pmatrix}b_2 c_3 - b_3 c_2\\b_3 c_1 -b_1 c_3 \\b_1 c_2 - b_2 c_1\end{pmatrix} \\ &= a_1 (b_2 c_3 - b_3 c_2)+a_2 (b_3 c_1 -b_1 c_3)+a_3 (b_1 c_2 - b_2 c_1) \\ &= a_1 b_2 c_3 - a_1 b_3 c_2 +a_2 b_3 c_1 -a_2 b_1 c_3 +a_3 b_1 c_2 -a_3 b_2 c_1 \\ &= det(A) \end{aligned} \end{equation}\]
.We can take the rows as vectors
\[\mathbf{u} = \begin{pmatrix}a_1\\b_1\\c_1\end{pmatrix} , \: \mathbf{v} = \begin{pmatrix}a_2\\b_2\\c_2\end{pmatrix} , \: \mathbf{w} = \begin{pmatrix}a_3\\b_3\\c_3\end{pmatrix}\]
.The vector triple product of the rows is
\[\begin{equation} \begin{aligned} \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) &= \begin{pmatrix}a_1\\b_1\\c_1\end{pmatrix} \cdot \begin{pmatrix}b_2 c_3 - b_3 c_2\\a_3 c_2 -a_2 c_3 \\a_1 b_3 - a_3 b_2 \end{pmatrix} \\ &= a_1 (b_2 c_3 - b_3 c_2)+b_1 (a_3 c_2 -a_2 c_3)+c_1 (a_2 b_3 - a_3 b_2) \\ &= a_1 b_2 c_3 - a_1 b_3 c_2 +a_3 b_1 c_2 -a_2 b_1 c_3 +a_2 b_3 c_1 - a_3 b_2 c_1 \\ &= det(A) \end{aligned} \end{equation}\]
.
