Question

How do you multiply `((0, 1, 0), (2, -1, 1), (0, 2, -1))` with `((-1, 2, 0), (4, 6, 0), (1, 0, 1))`?

Answer 1

Multiply `"row-line " xx " column-line of A"and " B` to get C:
`C = ((4,6,0 ), (-5,-2,1 ), (7,12,-1 ))`

Explanation:

Let `A = ((0,1,0 ), (2,-1,1 ), (0,2,−1 ))` and `B = ((−1,2,0 ), (4,6,0), (1,0,1 ))` then the product of `A*B = C` is given by:

`cij = a_(i1)b_(1j) + a_(i2)b_(2j) + cdots + a_(1n)b_(1n)`
`cij = sum_(k=1)^na_(ik)b_(kj)`
Now we have n = 3
`cij = sum_(k=color(red)(1))^color(red)3a_(ik)b_(kj)`
`c_(color(red)(1)color(blue)(1)) = a_(color(red)(1)1)b_(1color(blue)(1)` `+a_(color(red)(1)2)b_(2color(blue)(1) +` `a_(color(red)(1)3)b_(3color(blue)(1)`

`c_(color(red)(1)color(blue)(2)) = a_(color(red)(1)1)b_(1color(blue)(2)` `+a_(color(red)(1)2)b_(2color(blue)(2) +` `a_(color(red)(1)3)b_(3color(blue)(2)`

`cdots`

Note: each entry is made by three term obtained by multiplying row-line by column- line. That is:
row line = `((color(red)0, color(blue)1, color(green)0))` `((color(red)(-1)), (color(blue)(2)), (color(green)(0)) )`
`color(red)("red"xx"red")` + `color(blue)("blue"xx"blue")` + `color(green)("green"xx"green")`

Using this approach we find:

`C = ((4,6,0 ), (-5,-2,1 ), (7,12,-1 ))`