Video: Squaring and Cubing a Matrix

Take the matrix 𝐴 = [1, 0, 0 and 0, 5, 0 and 0, 0, 2]. Find 𝐴² and 𝐴³.

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Video Transcript

Take the matrix 𝐴 equals one, zero, zero, zero, five, zero, zero, zero, two. Then, what we need to do is find 𝐴 squared and then use this to find 𝐴 cubed.

Well, what 𝐴 squared is gonna be is matrix one, zero, zero, zero, five, zero, zero, zero, two multiplied by the matrix one, zero, zero, zero, five, zero, zero, zero, two. And we know that they can be multiplied by each other because the number of columns in matrix one is the same as the number of rows in matrix two. And that’s because they are both, in fact, three-by-three matrices. And therefore, our result is also gonna be a three-by-three matrix.

Now, in order to calculate what the first element in our matrix is going to be, what we do is we multiply the corresponding elements from the first row of our first matrix and the first column of our second matrix and then add them together. So we have one multiplied by one plus zero multiplied by zero plus zero multiplied by zero.

So then, from the next element along, what we do is this time look at the second column in our second matrix. So this time, what we’ll have is one multiplied by zero, zero multiplied by five, and zero multiplied by zero. So then finally, for the last element in our first row, what we’re gonna have is one multiplied by zero, zero multiplied by zero, and zero multiplied by two. And we got that because we moved on now to the third column in our second matrix.

Okay, great. So we now have the process. We use that just to fill in the rest of the matrix. So when we move to the next row down, what we do is we just move the row down in the first matrix and then we restart at the beginning with the first column in the second matrix. So it’s zero multiplied by one plus five multiplied by zero plus zero multiplied by zero. So then, we just continue our pattern and use this to complete our matrix. So we’ve done that here. So we’ve now got all of the calculations in place.

So now, what we need to do is calculate each of the elements of our matrix 𝐴 squared. Well, our first element is one cause we got one plus zero plus zero, then zero, then zero again. So that’s our first row complete. And then for our second row, we’re gonna get zero, 25, and zero. So then, for our final row, we’re gonna get zero, zero, four. So great, we found our matrix for 𝐴 squared. And that is one, zero, zero, zero, 25, zero, zero, zero, four.

So now, what we’re gonna do is move on to 𝐴 cubed. Well, in order to find 𝐴 cubed, what this is gonna be equal to is the matrix for 𝐴 squared multiplied by the matrix for 𝐴. So what it’s gonna be is the matrix one, zero, zero, zero, 25, zero, zero, zero, four multiplied by the matrix one, zero, zero, zero, five, zero, zero, zero, two. And we do that the same way that we multiplied two matrices earlier.

So to quickly remind us, we have a look at the first element. So what we do is we multiply the corresponding elements from the first row of the first matrix and the first column of the second matrix and add them together. So what we get is one multiplied by one add zero multiplied by zero add zero multiplied by zero. So then we use this method to carry on to complete the rest of our matrix. So we see here that we’ve written out all the calculations that we need. So our first row of the matrix 𝐴 cubed is gonna be one, zero, zero. The next row will be zero, 125, zero. So that will give us a final row of zero, zero, eight.

So great, we could solve the problem because we can say that if we take the matrix 𝐴, which is one, zero, zero, zero, five, zero, zero, zero, two, then 𝐴 squared will be one, zero, zero, zero, 25, zero, zero, zero, four. And 𝐴 cubed will be one, zero, zero, zero, 125, zero, zero, zero, eight.

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