Question

How do you multiply `((-2, -5, 3), (3, -1, 2), (1, 4, -2))` with `((1, 4, 3), (-3, -3, 2), (-2, -1, -2))`?

Answer 1

The formal definition of matrix multiplication is:

`c_(ik) = sum_(i,j,k) a_(ij)b_(jk)`

where `a,b,c` are entries in the matrices `A,B,C`, respectively, and `AB = C`. That means if `A` is `I xx J` and `B` is `J xx K`, then `C` is `I xx K`.

Let the first and second indices indicate the row and column, respectively, for matrices `A` and `B` (i.e., we are in row major).

If you take the sum of the dot products of a GIVEN row `bb(i)` in `bb(A)` with each column in `bb(B)`, for EACH `i`, that's matrix multiplication.

That means:

1. Multiply row 1 in `A` with column 1 in `B`. This entry goes in `c_(11)`.
2. Multiply row 1 in `A` with column 2 in `B`. This entry goes in `c_(12)`.
3. Multiply row 1 in `A` with column 3 in `B`. This entry goes in `c_(13)`.
4. Multiply row 2 in `A` with column 1 in `B`. This entry goes in `c_(21)`.
5. Multiply row 2 in `A` with column 2 in `B`. This entry goes in `c_(22)`.
6. Multiply row 2 in `A` with column 3 in `B`. This entry goes in `c_(23)`.
7. Multiply row 3 in `A` with column 1 in `B`. This entry goes in `c_(31)`.
8. Multiply row 3 in `A` with column 2 in `B`. This entry goes in `c_(32)`.
9. Multiply row 3 in `A` with column 3 in `B`. This entry goes in `c_(33)`.

This gives another `3xx3` matrix `C`:

`color(blue)(C) = AB`

`= [(-2, -5, 3), (3, -1, 2), (1, 4, -2)]xx[(1, 4, 3), (-3, -3, 2), (-2, -1, -2)]`

`[(sum_j a_(1j)b_(j1),sum_j a_(1j)b_(j2),sum_j a_(1j)b_(j3)),(sum_j a_(2j)b_(j1),sum_j a_(2j)b_(j2),sum_j a_(2j)b_(j3)),(sum_j a_(3j)b_(j1),sum_j a_(3j)b_(j2),sum_j a_(3j)b_(j3))]`

`[(-2*1 - 5*-3 - 3*2,-2*4-5*-3-1*3,-2*3-5*2-3*2),(3*1-1*-3-2*2,3*4-1*-3-1*2,3*3-1*2-2*2),(1*1-4*3-2*-2,1*4-3*4-1*-2,1*3+2*4-2*-2)]`

`[(-2 + 15 - 6,-8 + 15 - 3,-6 - 10 - 6),(3 + 3 - 4,12 + 3 - 2,9-2-4),(1 - 12 + 4,4 - 12 + 2,3 + 8 + 4)]`

`color(blue)([(7,4,-22),(2,13,3),(-7,-6,15)])`