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Matrix Multiplication - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Multiplying some matrices. So for this particular problem we are looking at matrix C and D and we want to multiply them together. So let’s look first at C times D.

So first thing we need to check is if these matrices are compatible if we can multiply them together. So matrix C is a two by two matrix matrix D is a two by two matrix, so we need to look and say okay these inner numbers match up so we can multiply them and we are going to be left with a two by two matrix. So we can multiply this and to make my life a little bit easier I’m going just to rewrite this next to each other. 4, 0, 1, 8 and then 0, -2, 2 and 3.

So we are multiplying these together and we are going to end up with a two by two matrix. So I always write it in, draw my little lines so I sort of know what exactly I am hoping to fill. Let’s figure this out.

So remember matrix multiplication is row by column. So here we are dealing with our first row and first column so this particular one is going to be dictated by this row and this column and then we multiply going down the row and we add up our answers. So this one is just going to be 4 times 0 and then we add that to 0 times 2. 4 times 0 plus 0 times 2 so this is all just going to end up being 0. We go to our next part. We are now dealing with the first row second column, so we are still dealing with the first row over here but now we’ve switched over to this column.

So that’s going to be 4 times -2 plus let's make this a little bit bigger then we have that, 4 times -2 is our first element and then we also have our second, 4 times -2 0 times 3. 0 times 3 is 0 so this is just turns into -8. Continuing on, so now this spot is the second row first column so now we are going to look at this second row first column multiplying and adding. 1 times 0 plus 8 times 2, so that turns out to be 16 and lastly second row second column 1 times -2 plus 8 times 3, 8 times 3 is 24 1 times -2 is -2, so this turns into 22 rewriting our answer to make it a little bit nicer and equals 0 -8 16 and 22.

So using our row operations we would try our rules in multiplication you're able to find the product of C times D. I do want to take a look at the opposite so I’m going to take a look at D times C as well.

So let’s go back over here. So D times C, so first we did the CD now let's switch the order. Our dimensions are still the same that hasn’t changed. We're dealing with two by two’s shows us we can multiply them because those inner numbers are the same and we are going to be left with a two by two matrix once again. To make our life easier go back up here and I’m just going to rewrite them next to each other so we see what is going on. 0, -2, 2, 3 times C 4, 0, 1 and 8 and we know once again we are going to be left with a let’s erase some of our scratch from upper figure and we know we're going to be left with a two by two. So for this one I’m not going to write out everything we are going to do some of the math in our head and just record it as we go.

This first entry is the row of the first and the column of the second. We have row times column and if you have different colors, if you have some sort of the way they have noted it they can always help you take some practice so you get used to how these all work together.

This one we just go 0 times 4, 0 plus -1 times sorry -2 times 1 -2 so this part is just -2. First row second column so we go to first row second column 0 times 0 -2 times 8 -16. Moving on second row first column, so this is our second row first column, two times 4 is 8, plus 3 times 1, 3, 8 plus 3 is 11. Last entry second row second column, 2 times 0 is 0, plus 3 times 8 24. Okay so we found D times C.

Last thing I want to point out is we have our C times D up here at this one right here and our D times C over here and what do you notice?. Numbers are different okay. So order of multiplication matters when you are dealing with the matrices normally we used to say 4 times 3 is the same thing as 3 times 4 with matrices that order actually does matter. So using the law of matrices making sure our order is right making sure our matrices are compatible to be multiplied we can multiply these two up.

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