Video: Powers of Square Matrices

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

04:29

Video Transcript

Consider the matrix 𝐴 equals one, one, two; one, zero, one; two, one, zero.

First part: find 𝐴 squared. Second part: find 𝐴 cubed.

So the first thing we need to do when we’re gonna solve this problem is consider what does 𝐴 squared mean. Well, what it means is matrix 𝐴 multiplied by matrix 𝐴. So now to work this out, what we’re gonna do is consider how we multiply a matrix with another matrix. Well, to work out the first term, what we do is we multiply the first row by the first column. So first of all, we multiply the first term in the first row by the first term in the first column, so one by one. And then you add the second term in the first row multiplied by the second term in the first column.

And then finally, you’ve got the last term in the first row multiplied by the last term in the first column. And when you add all of these together, you get one plus one plus four, which gives us six. So then to find the second term along the top, what we’re gonna do is the same process once more, but this time with the first row of the first matrix and the second column of the second matrix. So you have one multiplied by one plus one multiplied by zero plus two multiplied by one. And then finally for the last term across the top row, we have the first row and the third column. And we do the same as before, one multiplied by two plus one multiplied by one plus two multiplied by zero. So that’s the first row dealt with.

And then we move down to the second row. And for the second row, we deal with the second row of the first matrix and then first of all the first column of the second matrix. So we get one multiplied by one plus zero multiplied by one plus one multiplied by two. Then we carry on this pattern for the rest of this row. And then finally we move on to bottom row. And then finally at the bottom row, we just carry on the pattern as we’ve done before.

So, for instance, for the first term, it’s the bottom row of the first matrix multiplied by the first column of the second matrix, and so on and so forth. So then we add them up to find the terms in our 𝐴 squared. So our first term in the matrix is gonna be six cause we had one add one add four, which is six. And then our second and third terms are gonna be three and three because we add one add zero add two and two add one add zero. So then for our second row we’re gonna get three, two, and two. And then finally, on the bottom row, three two and five. So we’ve now got our matrix which represents the matrix 𝐴 squared.

So now we’re gonna move on and find the matrix 𝐴 cubed. Now to do this, we don’t actually have to multiply the matrix 𝐴 three times because we can use the result from 𝐴 squared and then just multiply this by matrix 𝐴. So what we’re gonna have is the matrix six, three, three; three, two, two; three, two, five multiplied by the matrix one, one, two; one, zero, one; two, one, zero. And to work this out, we use the same method that we used for the first part of the question. So the first term is gonna be the first row of the first matrix multiplied by the first column of the second matrix. So we get six multiplied by one plus three multiplied by one plus three multiplied by two.

So then we complete the rest of the first row in the same manner. And then finally we do the same for the bottom two rows. So now we’ve got all of the calculations needed to find the matrix of 𝐴 cubed, so the matrix 𝐴 cubed. So then when we calculate all the values, we’re left with the matrix 15, nine, 15; nine, five, eight; 15, eight, eight. So therefore, we can say that if we consider the matrix 𝐴, which is equal to one, one, two; one, zero, one; two, one, zero, then 𝐴 squared will be six, three, three; three, two, two; three, two, five as a matrix. And 𝐴 cubed will be the matrix 15, nine, 15; nine, five, eight; 15, eight, eight.

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