Question Video: Operations on Matrices Mathematics

Consider the matrices 𝐴 = [1 and 1 and 1]. 𝐴^(𝑇) = [1, 1, 1], 𝐡 = [π‘Ž, 𝑏, 𝑐], 𝐡^(𝑇) = [π‘Ž and 𝑏 and 𝑐]. Find 𝐴𝐡. Find 𝐡^(𝑇)𝐴^(𝑇).

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Video Transcript

Consider the matrices matrix 𝐴 is the three-by-one matrix one, one, one. The transpose of matrix 𝐴 is the one-by-three matrix one, one, one. Matrix 𝐡 is the one-by-three matrix π‘Ž, 𝑏, 𝑐, and the transpose of matrix 𝐡 is the three-by-one matrix π‘Ž, 𝑏, 𝑐. Then, find 𝐴𝐡 and find the transpose of 𝐡 multiplied by the transpose of 𝐴.

Well, now, if we consider 𝐴𝐡 first, then what we can see straightaway is that we can multiply the matrix 𝐴 by the matrix 𝐡. And we can do that because the number of columns in matrix 𝐴 is equal to the number of rows in matrix 𝐡. And what we also know is that our result matrix is going to be a three-by-three matrix. And that’s because there are three rows in matrix 𝐴 and three columns in matrix 𝐡. So therefore, 𝐴𝐡 is going to be equal to the three-by-one matrix one, one, one multiplied by the one-by-three matrix π‘Ž, 𝑏, 𝑐.

So now, to find the first row of our answer matrix, we’re going to multiply the first element of the first row in our first matrix by each of the elements in our other matrix. So we’re gonna have one multiplied by π‘Ž, one multiplied by 𝑏, and one multiplied by 𝑐. And then, due to the nature of the matrices that we’re looking at, our following two rows are going to be exactly the same. So we’re going to have one multiplied by π‘Ž, one multiplied by 𝑏, and one multiplied by 𝑐 for each of those two rows. And so this is gonna give us the three-by-three answer matrix π‘Ž, 𝑏, 𝑐, π‘Ž, 𝑏, 𝑐, π‘Ž, 𝑏, 𝑐.

So, great, we’ve now solved the first part of the question. What we’re going to do is move on to the second part. And what we’re gonna do here is multiply the transpose of matrix 𝐡 by the transpose of matrix 𝐴. So once again, we’re going to have a three-by-one matrix multiplied by a one-by-three matrix. And we know that they are going to multiply. And that’s because, again, we’ve got the same number of columns in the first matrix as the number of rows in the second matrix. And once again, we know that our answer matrix is going to be a three-by-three matrix. And that’s because the number of rows in our first matrix is three. And that’s the first matrix, which is the transpose of matrix 𝐡. And the number of columns in our second matrix, which is the transpose of 𝐴, is also three.

So what we can say is that the transpose of matrix 𝐡 multiplied by the transpose of matrix 𝐴 is going to be equal to the three-by-one matrix π‘Ž, 𝑏, 𝑐 multiplied by the one-by-three matrix one, one, one. So this time what we’re going to do once again is multiply the first element from the first row in the first matrix by each of the elements in our second matrix. So we’re gonna get π‘Ž multiplied by one, π‘Ž multiplied by one, and π‘Ž multiplied by one. And this is the first row of our answer matrix. Then our second row is going to be 𝑏 multiplied by one, 𝑏 multiplied by one, 𝑏 multiplied by one. And then the final row is going to be 𝑐 multiplied by one, 𝑐 multiplied by one, 𝑐 multiplied by one. And this is going to give us our three-by-three answer matrix π‘Ž, π‘Ž, π‘Ž, 𝑏, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐.

So we now solved both parts of the question. However, there’s something interesting that we might notice between the two answers. If we look at the answer to the first part, we’re gonna have the three-by-three matrix π‘Ž, 𝑏, 𝑐, π‘Ž, 𝑏, 𝑐, π‘Ž, 𝑏, 𝑐. Well, then, if we take the transpose of this matrix. And what the transpose means is that we switch the corresponding rows and columns. And for a square matrix, this is the same as swapping elements over the main diagonal. And if we do this, we’ll see that we’ll get the three-by-three matrix π‘Ž, π‘Ž, π‘Ž, 𝑏, 𝑏, 𝑏, 𝑐, 𝑐, 𝑐, which is in fact the answer we got when we multiplied together the transpose of matrix 𝐡 by the transpose of matrix 𝐴.

What we have in fact shown here is an example of the matrix property that states that the transpose of 𝐴𝐡 is equal to the transpose of 𝐡 multiplied by the transpose of 𝐴.

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