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 $M(Nv) = (MN)v$ ?

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Answer 1 · 2016-08-04T16:57:21

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

(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.

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