Cross product determinant's matrix The cross product $a \times b$ can be represented by the determinant
$$\mathbf{a}\times\mathbf{b}= \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
a_1 & a_2 & a_3 \\
b_1 & b_2 & b_3 \\
\end{vmatrix}.$$
Does the matrix whose determinant is this have any significance?
 A: You can let the matrix act by ordinary matrix multiplication on ordinary vectors in three-dimensional space.
This will transform a vector in a triple containing the original vector and the lengths of the two projections on $a$ and $b$.
While I feel that this counts as "any significance", it isn't very satisfactory, because the matrix, as you presented it, does not allow for matrix multiplication.
I think it is a much more useful point of view, to first view $i$,$j$,$k$ as three scalar variables (better denoted by $x$, $y$, $z$), then take the determinant of your matrix and then regard the cross product as the gradient vector this determinant.
A: You could think of the entries as all being Quaternions, i.e. the matrix is an element of $M_3(\mathbb{H})$. This gives you a genuine way to act on vectors and do matrix multiplication. But you need to be careful about converting between the two different interpretations $i=(1,0,0)\in\mathbb{R}^3$ versus $i\in\mathbb{H}$
https://en.m.wikipedia.org/wiki/Quaternion
