Question

Let M and N be matrices , `M = [(a, b),(c,d)] and N =[(e, f),(g, h)],` and `v` a vector `v = [(x), (y)].` Show that `M(Nv) = (MN)v`?

Answer 1

This is called an associative law of multiplication.
See the proof below.

Explanation:

(1) `Nv = [(e,f),(g,h)]*[(x),(y)] = [(ex+fy),(gx+hy)]`

(2) `M(Nv)= [(a,b),(c,d)]* [(ex+fy),(gx+hy)] = [(aex+afy+bgx+bhy),(cex+cfy+dgx+dhy)]`

(3) `MN= [(a,b),(c,d)]* [(e,f),(g,h)] = [(ae+bg,af+bh),(ce+dg,cf+dh)]`

(4) `(MN)v = [(ae+bg,af+bh),(ce+dg,cf+dh)]*[(x),(y)] = [(aex+bgx+afy+bhy),(cex+dgx+cfy+dhy)]`

Notice that the final expression for vector in (2) is the same as the final expression for vector in (4), just the order of summation is changed.

End of proof.