Question

How do you multiply `((1, -4), (4, -1))` and `((3, -2), (0, -3))`?

Answer 1

Matrix multiplication basically involves multiplying each row of the left matrix by all the columns of the right matrix.

I could show you a general notation for this at the end of the answer, but an example is easier to understand.

Since an `MxxK` multiplied by a `KxxN` matrix generates an `MxxN` matrix, a `2xx2` multiplied by a `2xx2` gives a `\mathbf(2xx2)` matrix.

`color(blue)([(1,-4),(4,-1)][(3,-2),(0,-3)])`

`= [(1*3 + (-4*0),1*-2 + (-4*-3)),(4*3 + (-1*0),4*-2 + (-1*-3))]`

`= color(blue)([(3,10),(12,-5)])`

The general notation for that (for `2xx2` matrices for simplicity) is:

`[(color(blue)(a_(11)), color(blue)(a_(12))),(color(highlight)(a_(21)), color(highlight)(a_(22)))]xx[(color(orange)(b_(11)), color(red)(b_(12))),(color(orange)(b_(21)), color(red)(b_(22)))]`

`= [(color(blue)(a_(11))color(orange)(b_(11)) + color(blue)(a_(12))color(orange)(b_(21)), color(blue)(a_(11))color(red)(b_(12)) + color(blue)(a_(12))color(red)(b_(22))), (color(highlight)(a_(21))color(orange)(b_(11)) + color(highlight)(a_(22))color(orange)(b_(21)), color(highlight)(a_(21))color(red)(b_(12)) + color(highlight)(a_(22))color(red)(b_(22)))]`