Video: Using Operations on Matrices to Evaluate a Given Algebraic Expression Involving Identity Matrices

Given that 𝐴 = [−5, −6 and 5, 0], find 𝐴² + 5𝐴 + 30𝐼.

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Video Transcript

Given that the matrix 𝐴 equals negative five, negative six, five, zero, find 𝐴 squared plus five 𝐴 plus 30𝐼.

So, in order to solve this problem, what we’re going to do is deal with each part separately, so our 𝐴 squared, our five 𝐴, and our 30𝐼. Well, the first thing which we’re just gonna quickly have a look at is 𝐼 because you might think, “Well, what is 𝐼?” 𝐼 is the identity matrix. And the identity matrix is a matrix that is a square matrix; it has ones on the main diagonal and zeros elsewhere. Because we’re looking at two-by-two matrices, that would give us the identity matrix one, zero, zero, one.

Okay, so, we’ve now seen what 𝐼 is, so our identity matrix. Now, let’s start with 𝐴 squared where we’re going to find 𝐴 squared by multiplying the matrix negative five, negative six, five, zero by the matrix negative five, negative six, five, zero. First of all, we know that our answer matrix is gonna be a two-by-two matrix. And that’s ’cause if we’re going to multiply a two-by-two matrix by a two-by-two matrix, then, as we said, the answer matrix would also be a two-by-two matrix.

Now, in order to find out first element, what we’re going to do is multiply the first element in the first row of the first matrix by the first element in the first column of the second matrix. And then, we’re gonna add this to the second element in the first row of the first matrix multiplied by this second element in the first column of the second matrix, which gives us negative five multiplied by negative five plus negative six multiplied by five, which is gonna give us 25 minus 30. Then for our next element, what we do is multiply the corresponding elements from the first row in the first matrix by the second row in the second matrix and then add them together. And this gives us negative five multiplied by negative six plus negative six multiplied by zero, which is gonna give us 30 plus zero.

Then repeating this method for the next element, we have five multiplied by negative five plus zero multiplied by five, which gives us negative 25 plus zero. And then finally, we have five multiplied by negative six plus zero multiplied by zero, which gives us negative 30 plus zero. Now, we can simplify this. And when we do, we get the matrix negative five, 30, negative 25, negative 30. Okay, great. That’s 𝐴 squared dealt with.

Now, what we’re going to do is take a look at five 𝐴. So now, to find five 𝐴, this is even more straightforward because what we need to do here when we’re multiplying a scalar by a matrix is just multiply each element within our matrix by the scalar value, in this case, five. So, we’re gonna have five multiplied by the matrix negative five, negative six, five, zero. And when we do that, we’re going to get the matrix negative 25, negative 30, 25, zero.

Now moving on, what we want to do is calculate 30𝐼. And again, what this is is a scalar value multiplied by a matrix, so 30 multiplied by the matrix one, zero, zero, one, which is our identity matrix, which is going to give us the matrix 30, zero, zero, 30. Okay, great. So now, what we need to do is to put everything together to find 𝐴 squared plus five 𝐴 plus 30𝐼. So now, if we take a look at 𝐴 squared plus five 𝐴 plus 30𝐼, this is gonna be equal to the matrix negative five, 30, negative 25, negative 30 plus the matrix negative 25, negative 30, 25, zero plus the matrix 30, zero, zero, 30.

So now if we’re looking to add our matrices, what we need to do is, in fact, add together the corresponding elements from each of our matrices. So, our first element is going to be negative five plus negative 25 plus 30. Then, our next element is gonna be 30 plus negative 30 plus zero. Then carrying on in the same way, our next two elements are going to be negative 25 plus 25 plus zero and negative 30 plus zero plus 30.

Now, what we need to do is calculate each of our individual elements. So, first of all, negative five plus negative 25 is negative 30 add 30 is zero. Then, if we move on to the next element, 30 add negative 30 is zero plus zero is still zero. Then moving to the bottom row, we have negative 25 plus 25 which is zero plus zero is zero and then, finally, negative 30 plus zero plus 30 which is zero as well.

So therefore, we can say that given that matrix 𝐴 is equal to negative five, negative six, five, zero, then 𝐴 squared plus five 𝐴 plus 30𝐼 is the matrix zero, zero, zero, zero.

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