Evaluate the matrix product matrix eight, 11, negative three, seven multiplied by the matrix 10, negative one, three, one.
So first of all, let’s remind ourselves about multiplying matrices. The first thing we note is that when we’re multiplying matrices together, if we look at their dimensions, then the second dimension of the first matrix must be equal to the first dimension of the second matrix. So what does this mean in practice? Well, in our question, we’ve got a two-by-two matrix multiplied by a two-by-two matrix, so therefore we know that the two and the two are the same. So we know that they can be multiplied together. And then we know that the remaining dimensions are gonna give us the dimensions of our result. So in our example here, our result of multiplying an 𝑚-by-𝑛 matrix and an 𝑛-by-𝑝 matrix would give us an 𝑚-by-𝑝 matrix.
Well, if we use this and apply it to what we’re looking at, well, we could see that the remaining two dimensions are twos. So we’d have a two-by-two matrix as our result. And actually, this is kind of what we would think because if we’re looking at a two-by-two multiplied by a two-by-two, it makes sense that the result will also be a two-by-two matrix.
Now to multiply together the matrices, the first element is gonna be formed by multiplying together the corresponding elements of the first row of the first matrix and the first column of the second matrix and then adding them together. So the first element is gonna be eight multiplied by 10 plus 11 multiplied by three. And then the second element will involve us moving along to the second column of the second matrix. So we’ve got eight multiplied by negative one plus 11 multiplied by one.
And then we move to the next row in the first matrix. So we’ve got negative three multiplied by 10 plus seven multiplied by three. And then for the final element, we have negative three multiplied by negative one plus seven multiplied by one. So great, now all we need to do is work out the values for each of our elements.
So for the first element, we’re gonna get 113 because that’s 80 plus 33. And then we’ve got negative eight plus 11, which gives us three, then negative nine because this is negative 30 plus 21. So then our final element is 10. So we can say that the answer to the matrix product is the two-by-two matrix 113, three, negative nine, 10.