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Matrix Multiplication - Concept

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!

When working with matrices, we can perform a number of matrix operations including matrix multiplication. When multiplying matrices, we first need to ensure that the matrices have the same dimensions, which is the number of rows times the number of columns. The resulting matrix after multiplication has the dimensions of the outer two dimensions. Each value is equal to the product of the corresponding row and column.

Matrix Multiplication is one of the trickier things that we do in Matrices. And so we're going to take a look at sort of an abstract idea and sort of figure out how it all works together.
When multiplying Matrices the first thing we need to do is make sure we actually can multiply matrices together, and not all matrices can always be multiplied together.
Okay so looking at it well the first thing we need to do is look at the dimensions, remember dimensions are rows by columns so this first matrices is two rows and two columns so this is a 2 by 2 matrice here. The rows by columns two rows two columns so this is a two by two as well.
Okay in order to multiply matrices these two have to be equal, the two inside the outer, the second dimension of your first matrix the first dimension of your second. If these are equal you can multiply and more so your resulting matrix the matrix you're going to be left with is going to have dimensions of your outer most two. So this is going to be your resulting matrix dimensions. So in this case we have all two's so everything is equal, tells us that we're going to be able to multiply it and the resulting matrix is going to be a two by two matrix as well.
Okay I am going to write this matrix to be really big cause it will be fairly involved. So we're going to have a slot here slot here slide here and a slot here. So we have a fairly big two by two. And how this works is to find this first spot, the first matrix is this element is determined by the row of the first matrix same as and the column as the second. So it's the same as your dimensions row by column this is in the first row first column. So it's determined by the first row and first column. So we have four things determining this one spot and how it works is you start at the first entry on your row and you start the first entry of your column and you multiply and you add as you go down. So this one we take a times w and we add that to b times y, so this is aw+by okay.
A little bit confusing but let's do another one to see if we have it, so for this entry right here it is in the first row second column. So again we're going to look at the first row over here but this time we have switched over to the second column. So again going down the row down the row and down the column we take 8 times x plus b times e okay.
Now these are pretty involved equations I wouldn't really recommend memorizing them but hopefully the concept is what you're looking at okay. The first matrix determines the row, second one determines the column. Let's say we want to jump forward and look at this entry right here, it's in the second row second column. So we need to go to our Matrices and our second row second column, again the first matrix determines the row the second one determines the column. So then we just go down the way cx+dz, we just finish this one up cause we only one more entry to go.
First row, sorry that's our first row, our second row first column so we're dealing with the second row of our first matrix first column of the second this will be cw times dy uh sorry cw+dy. So when we're multiplying matrices always make sure your dimensions are compatible it's always going to be if your innermost dimensions are the same your outermost dimensions are going to be your resulting matrix and then whatever spot you're looking for the row is determined by the first matrix the columns are determined by the second.

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