Question

How do you multiply `((0, 1, 0), (6, -3, 1), (-1, 4, -2)))` and `((6, 7, 1), (2, 10, 5), (1, -10, 9))`?

Answer 1

The answer is `=((2,10,5),(31,2,0),(0,53,1))`

Explanation:

The multiplication of 2 matrices is

`((a,b,c),(d,e,f),(g,h,i))*((k,l,m),(n,p,q),(r,s,t))`

`((ak+bn+cr,al+bp+cs,am+bq+ct),(dk++en+f r,dl+ep+fs,dm+eq+ft),(gk+hn+ir,gl+hp+is,gm+hq+it))`

`((0,1,0),(6,-3,1),(-1,4,-2))*((6,7,1),(2,10,5),(1,-10,9))`

`=((2,10,5),(31,2,0),(0,53,1))`

Answer 2

There is a formula, but many find the following process easier to remember.

Explanation:

`((0,1,0),(6,-3, 1),(-1,4,-2)) ((6,7,1),(2,10, 5),(1,-10,9))`

Find the first row of the product

Take the first row of `((0,1,0),(6,-3, 1),(-1,4,-2))`, and make it vertical. (We'll do the same for the second row in a minute.)

`{: (0),(1),(0) :} ((6,7,1),(2,10, 5),(1,-10,9))`

Now multiply times the first column and add to get the first number in the first row of the answer:
`(0 xx 6) + (1 xx 2)+ (0 xx 1) = 0+2 + 0 = 2`

Next multiply times the second column and add to get the second number in the first row of the answer:
`(0 xx 7) + (1 xx 10)+ (0 xx -10) = 0+10 + 0 = 10`

Finally, multiply times the third column and add to get the third number in the first row of the answer:
`(0 xx 1) + (1 xx 5)+ (0 xx 9) = 0+5 + 0 = 5`

At this point we know that the product looks like:

`((0,1,0),(6,-3, 1),(-1,4,-2)) ((6,7,1),(2,10, 5),(1,-10,9)) = ((2,10,5),("-","-","-"),("-","-","-"))`

Find the second row of the product
Find the second row of the product by the same process using the second row of `((0,1,0),(6,-3, 1),(-1,4,-2))`

Stand up the second row, then multiply and add to find the numbers in the second row of the answer.
`{: (6),(-3),(1) :} ((6,7,1),(2,10, 5),(1,-10,9))`

First number in the second row of the answer:
`(6 xx 6) + (-3 xx 2) + (1 xx 1) = 36 -6 + 1 = 31`

Second number in the second row of the answer:
`(6 xx 7) + (-3 xx 10) + (1 xx -10) = 42-30-10 = 2`

Third number in the second row of the answer:
`(6 xx 1) + (-3 xx 5) + (1 xx 9) = 6-15+9 = 0`

At this point we know that the product looks like:

`((0,1,0),(6,-3, 1),(-1,4,-2)) ((6,7,1),(2,10, 5),(1,-10,9)) = ((2,10,5),(31,2,0),("-","-","-"))`

Find the third row of the product.

`{: (-1),(4),(-2) :} ((6,7,1),(2,10, 5),(1,-10,9))` to get:

`-6+8-2=0`, then `-7+40+20 = 53`, then `-1+20-18 = 1`

Write the answer

`((0,1,0),(6,-3, 1),(-1,4,-2)) ((6,7,1),(2,10, 5),(1,-10,9)) = ((2,10,5),(31,2,0),(0,53,1))`