Video: Polynomials of Square Matrices

Consider the matrices 𝐴 = [1, 2 and −3, −4], 𝐼 = [1, 0 and 0, 1]. Find 𝐴², and 𝐴² + 3𝐴 + 2𝐼.

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Video Transcript

Consider the matrices 𝐴 is equal to one, two, negative three, negative four; 𝐼 is equal to one, zero, zero, one. Find 𝐴 squared. And then find 𝐴 squared plus three 𝐴 plus two 𝐼.

So first of all, what we’re gonna do is start with 𝐴 squared. What this means is matrix 𝐴 multiplied by matrix 𝐴. So we’ve got one, two, negative three, negative four multiplied by one, two, negative three, negative four. So if we want to look at the first term in our answer matrix, then we’re gonna find this using the first row from first matrix and the first column from the second matrix.

So the way that we use these to find the first term is by multiplying the corresponding terms. So, for instance, we multiply the first term in the first row by the first term in the first column. So it’s one multiplied by one. And then we add it to the second term in the first row multiplied by the second term in the first column, so two multiplied by negative three.

So now, for the second term in the first row, what we’re gonna do is multiply the corresponding values from the first row in the first matrix and the second column in the second matrix. So we have one multiplied by two plus two multiplied by negative four. And then we carry this pattern on for the bottom row. So we’ve next got the second row in the first matrix by the first column in the second matrix. So we have negative three multiplied by one add negative four multiplied by negative three. And then finally, the last term is gonna be negative three multiplied by two plus negative four multiplied by negative four.

Okay, great, so now let’s work out our values in our answer matrix. So, therefore, we can say that if we square matrix 𝐴, we can get the matrix negative five, negative six, nine, 10. So now, we can move on to the second part of the question where we need to find 𝐴 squared plus three 𝐴 plus two 𝐼. Well, the first thing I need to do is work out three 𝐴 because we know what 𝐴 squared is because we got that in the first part of the question. So now, we need to work out three 𝐴. What this means is three multiplied by the matrix 𝐴.

And to calculate this, all we do is multiply each of the terms by three. So we had three multiplied by one, three multiplied by two, three multiplied by negative three, and three multiplied by negative four. And when we do that we get the matrix three, six, negative nine, negative 12. And then we do the same for two 𝐼. So this time it’s two multiplied by the matrix 𝐼 which is one, zero, zero, one. And this again give us the matrix two, zero, zero, two. And that’s because we had two multiplied by one, two multiplied by zero, two multiplied by zero, and then two multiplied by one.

Okay, great, so now we have all of the parts we need to find 𝐴 squared plus three 𝐴 plus two 𝐼. So what we’re gonna get is negative five, negative six, nine, 10 — that’s because that’s our 𝐴 squared — then plus three 𝐴 which is three, six, negative nine, negative 12; then finally plus two 𝐼 which is two, zero, zero, two. So then to work this out, what we do is we add the corresponding values.

So for the top-left value, we’re gonna have negative five add three add two. And then the top-right value, we’re gonna have negative six add six add zero, bottom left, nine add negative nine add zero, and then finally, bottom right, 10 add negative 12 add two. And, therefore, this gives us the matrix zero, zero, zero, zero. And we got that because we had negative five add three add two. So negative five add five is zero. Then we had negative six add six which is zero. Then we had nine add negative nine which is zero. And then finally, 10 add negative 12 add another two which is also zero.

So we can say that if we consider the matrices 𝐴 equals one, two, negative three, negative four and 𝐼 equals one, zero, zero, one, then 𝐴 squared is gonna be the matrix negative five, negative six, nine, 10. And 𝐴 squared plus three [𝐴] plus two 𝐼 is gonna be the matrix zero, zero, zero, zero.

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