How to Multiply Matrices
Question
Matrices represent important operations called
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Hi, my name is Bassem Saad. I'm a Math Ph.D. candidate at U.C. Davis, and I'm here today for About.com to show you how to multiply matrices.The first step in multiplying matrices is to check and see that the inner-dimensions are the same; that is, that the number of columns in the left-most matrix equals the number of rows in the right-most matrix.Let's multiply this one-by-two matrix to this two-by-two matrix. The outer dimensions tells us that we're going to get a one-by-two matrix.To find the first element inside this one-by-two matrix -- that is, the element inside the first row, first column -- we'll have to multiply the first row of the first matrix, times the first column of the second matrix.Now the way we do that is we multiply their respective elements and then we take the sum. So, we multiply one times minus one, we add that to four times five, and we get 19. That is 19 belongs in the one, one position of our product.To find the element inside the first row, second column, we multiply the first row of the first matrix to the second column of the second matrix. So we have one times zero, plus four times two, which equals eight. We put eight inside the one, two position. Now we have our product.Let's take a look at another example.Here we have a two-by-two matrix, times a two-by-three matrix. Our outer dimensions tells us we're going to get a two-by-three matrix. And to find the first element inside the two-by-three matrix – that is, the element inside the first row and first column – we're going to have to multiply the first row of the first matrix, times the first column of the second matrix.So that is, three times minus one, plus minus one times four, and that's going to give us a minus seven. So minus seven goes into the one, one position. To find the next element – that is, the element inside the first row, second column, we want to multiply the first row times the second column. I'll do that very quickly in my head to give us minus one.To find the element inside the first row, third column, we want to multiply the first row, third column; that'll give us the element, six. To find the element inside the second row, first column, we want to multiply the second row times the first column; that'll give us 20. To find the element inside the second row, second column, we want to multiply the second row times the second column, which is five. And for the last one, we want to multiply the last row times the last column, which will be zero.Now we have our product. Matrices represent important operations called linear transformations. These transformations are useful in modeling all sorts of interesting observations in physics. Being able to multiply matrices is a crucial step in understanding these observations.Thanks for watching, and to learn more, visit us on the web at About.com.