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linear algebra: motivation for matrix multiplication ?

I have been studying linear algebra out of a proof-based text, and it has thus far offered much insight into matrices, such as the fundamentals upon which determinants have been established. The only problem that I have is that it has not given me a reason for WHY matrix multiplication is defined the way that it is. I would like to know the motivation behind defining matrix multiplication in the way that it is defined.

1 Answer

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  • JB
    Lv 7
    1 decade ago
    Favorite Answer

    It is natural to describe the function F:R^2 ->R^2 that carries (x,y) to (ax+by, cx+dy) by the matrix

    [ a b ]

    [ c d ]

    and the function G:R^2 ->R^2 that carries (x,y) to (ex + fy, gx + hy) by the matrix:

    [ e f ]

    [ g h ]

    What would the matrix for the composition (GoF)R^2 -> R^2, defined by

    (GoF)(x,y) = G(F(x,y))? Well, it would be nice to denote it symbolically by

    [ e f ] [ a b ]

    [ g h] [ c d ] ..... ..... ..... ...(*)

    But you can do the algebra and figure out that

    G(F(x,y)) = G((ax+by, cx+dy)) = (e(ax+by) + f(cx+dy), g(ax+by) + h(cx+dy)) = ((ea+fc)x + (eb+fd)y, (ga+hc)x + (gb+hd)y)

    But the matrix of that is

    [ (ea+fc) (eb+fd) ]

    [ (ga+hc) (gb+hd) ] ..... ..... ..... ..... (**)

    So we better define the product of the matices (*) above as this matrix (**).

    In short, matrix multiplication is defined the way it is so that matrix multiplication will work consistently with composition of linear transformations.

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