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Matrix Multiplication - Problem 2
Multiplying matrices. In this example we are going to multiply matrices that have quite different appearances. So we are looking at this A is this you know small fat matrice B is this taller matrice we are going to see if we can actually multiply some of these together.
The first thing I want to look at is A times B, see if we can multiply this up. When we are multiplying matrices we need to make sure that the first row dimensions match up so that we can multiply them. So dimensions of A is dimensions are rows by columns so this is a one by three matrix dimensions of B rows by columns is a three by two.
In order to be able to multiply matrices are inner dimensions have to match up in this case they are both three, so that works and our outer dimensions are going to be the result we are left with, so in this case we are going to be left with A one by two okay. So I know that I can multiply these so I’m going to rewrite them next to each other so we can multiply them up.
This is going to be 2, 1, 8, so that 1 in the middle so it doesn’t look like an 18 and 1, 4, 3, 0, -2, 5 okay, and we said that we were going to have a one by two matrix, so that tells us we have one row and just two columns. So it's just going to be a matrice with two elements. That’s perfectly fine.
So how we multiply these together is to find this first one it’s in the first row first column so the row of this determines the row and the column of this would determine the column, so we basically are going to take the only row we have times this first column and then just add as we go. So 2 times 1 is 2, 1 times 4 is 4, 8 times 3 is 24 add those all up together and we end up with 30. Going over here this is in the first row second column, so we deal with the row from our first matrix and the column from our second and multiply and add. 2 times 0 is going to give us 0, one times -2 is -2 8 times 5 is 40. Adding these all up together -2 plus 40 is 38.
Okay you can choose to write up all these little steps if you want, if you can do it in your head that’s great whichever is fine for you. So I made sure that our matrices are compatible to be multiplied using rules of multiplication we ended up with a one by two matrix with the answer 30 and 38.
So let’s go back and see if we can multiply the other way, B times A. okay matrix B rows by columns we already found to be three by two, matrix A is a one by three matrix and so looking seeing this our inner two numbers are not the same, so that tells us right here we can’t multiply these together okay. So no solution can’t do it. Quite easier than this one up here but we figured just by checking our dimensions we can multiply these together.
So it’s always important is to check your dimensions make sure your matrices are compatible and then using your rows times columns find your solution.