Question Video: Properties of Multiplying Matrices

Suppose 𝐴 = [1, −2, −3] and 𝐵 = [8 and 1 and −3]. Find the product 𝐴𝐵 and 𝐵𝐴.

04:31

Video Transcript

Suppose the matrix 𝐴 is the one-by three matrix one, negative two, negative three and the matrix 𝐵 is the three-by-one matrix eight, one, negative three. Find the product 𝐴𝐵 and find the product 𝐵𝐴.

So before we start to answer this question, let’s remind ourselves a little bit about matrix multiplication. If we’ve got the matrix 𝐴 multiplied by the matrix 𝐵, well then, first of all, what we can think about is if we have, for instance, an 𝑚-by-𝑛 matrix multiplied by an 𝑛-by-𝑝 matrix, to enable our multiplication to happen, then one thing must be true. And that is that the second dimension of our first matrix must be the same as the first dimension of our second matrix. So in this case, we can see we’ve got the shared 𝑛. And then what we also know is that the first dimension of the first matrix and the second dimension of the second matrix are gonna come together to give us the dimensions of our result. So therefore, our result in this example is gonna be an 𝑚-by-𝑝 matrix.

So now, we’re looking to solve our problem. We’re gonna first of all take a look at our matrices. We’ve got a one by three and the three by one. Well, to check that they can actually be multiplied, we check the second dimension of the first matrix and the first dimension of the second matrix, and they’re the same. So this is okay. Well, then, if we take a look at the product for 𝐴𝐵, then this is gonna be the remaining dimensions, so it’s gonna be one by one. And that’s because the product is 𝐴𝐵. So we have our 𝐴 matrix first and then our 𝐵 matrix. So we’ve got a one-by-one matrix as the result.

So now, to find out the value of our one-by-one matrix, what we’re gonna do is take a look at each of our elements. And then what we do is multiply the corresponding elements, so the first element in the first matrix by the first element in the second matrix. So we’re gonna have one multiplied by eight. And then we do the same for the second elements. So we have negative two multiplied by one, and we add this to the first. And then finally, we add the product of the third corresponding elements. So it’s gonna be negative three multiplied by negative three, which is gonna give us eight minus two plus nine, which is gonna give us our final answer for 𝐴𝐵, which is the one-by-one matrix 15. Okay, great! So now let’s move on to find the product of 𝐵𝐴.

Well, with 𝐵𝐴, we’ve actually got our matrices the other way round. So we’ve got three-by-one matrix multiplied by a one-by-three matrix. Well, here, once again, we can see that the second dimension of the first matrix is the same as the first dimension of the second matrix. So it means that we can multiply our matrices. And then the same as before, we can have a look at the first dimension of our first matrix and the second dimension of our second matrix. And this is gonna tell us the dimensions of our result, so it’s gonna be the three-by-three matrix. Okay, great.

So now let’s multiply our matrices and form our three-by-three matrix. Well, as the same as in the first part of the question, we’re gonna multiply corresponding elements. But this time, obviously, we need to make a three-by-three matrix. So the way it’s gonna work, first of all, is we’re gonna have our first element from our first column, or our only column, in our first matrix multiplied by the first element in the second matrix. So it’s gonna be eight multiplied by one. And that’s the first element of our answer matrix complete. Now we’re gonna move on to the next element.

Then the next element is found by multiplying eight by negative two because what we’ve done is move along to the second element in our second matrix and then finally multiply our eight by the final element in our second matrix. So we get eight multiplied by negative three. And then that’s the first row of our answer matrix complete. And then, for the second row, we just move down a row in our original matrix. So now we’re gonna have one, and then we’re gonna have one multiplied by one. And then we’ve got one multiplied by a negative two, etcetera. And then we continue this pattern on to finish our answer matrix. Okay, great! Now all we need to do is work out each of the individual elements. And when we do that, we’re gonna get the matrix eight, negative 16, negative 24, one, negative two, negative three, negative three, six, and nine.

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