Linear algebra: cross product property

Let the entries of $a$ be $a_1$ and $a_2$ and those of $b$ be $b_1$ and $b_2$. Then the left hand side reads

$$(a_1M_1 + a_2M_2) \times (b_1M_1 + b_2M_2).$$

Using the distributive property we know from ordinary multiplication, this expands to

$$a_1b_1 (M_1 \times M_1) + a_1b_2 (M_1 \times M_2) + a_2b_2 (M_2 \times M_2) + a_2b_1 (M_2 \times M_1)$$

Ignoring terms of the form $X \times X$ (because they are zero) and using the fact that $M_2 \times M_1 = - M_1 \times M_2$ we end up with:

$$(a_1b_2 - a_2b_1)(M_1 \times M_2)$$

This is already almost what we wanted: obviously the scalar $(a_1b_2 - a_2b_1)$ appearing here equals $\det[a, b]$ so all we need is the general fact that $\det A = \frac{1}{\det A^{-1}}$ that holds for all invertible matrices $A$.

This answers the question.

Notice that I tried to avoid actually computing any cross products and worked with their properties (bilinearity, anti-symmetry) instead. This is often a good idea when cross products are involved.