Given that 𝐴 and 𝐵 are these matrices, find 𝐴 plus 𝐵 times 𝐴.
So to follow the order of operations, we need to add 𝐴 and 𝐵 first and then that we will multiply by 𝐴. So in order to add these, we will add numbers in the corresponding spots. So for example, negative five and four are both on the top left-hand corner. Six and negative three are both on the first column, the bottom row. We will add negative five and six and then six plus five and now we simplify.
So we have the 𝐴 plus 𝐵 as negative one, one, negative nine, eleven. And now we need to take that matrix and multiply by 𝐴. So multiplying the matrix that we got from 𝐴 plus 𝐵, we’re now multiplying that by 𝐴. So multiplying matrices is different than adding. Before we begin multiplying, let’s decide how big our matrix should be.
These are both two-by-two matrices and the two inside numbers should be exactly the same; if they’re not, it won’t work. So here we do have that. And then the outside twos will be the size of the product, so the actual size of our final answer matrix. So just like with adding, we multiply the first two numbers together in the top left-hand corner, but we also add one times negative six. This will be our very first number. So we’ll have to add those together; that will be our first number. And it will be on the top left-hand corner of our final matrix.
Now below it, still in column number one, we take negative nine times negative five and then we add. Eleven times negative six, this will be our second number in column number one. So notice we’re taking the first two rows which are the only two rows in our product of 𝐴 plus 𝐵, that matrix, and we’re multiplying by the first column in 𝐴, this negative five, negative six.
So now to get our other numbers in column number two for our final answer, we will multiply by this column instead. So we take negative one times negative five and one times six. And now it’s the same thing, but for the second row. So we take negative nine times negative five and eleven times six and now we simplify.
So we went ahead and multiplied all the numbers together. And now to simplify, we’ll add each of these numbers together. Therefore, after multiplying by 𝐴, our final matrix would be negative one, 11 negative 21, 111. So again, we added 𝐴 and 𝐵 together and then we multiply it by 𝐴 and that gives us our final matrix.