 In this video, we present the solutions to question number nine from the practice midterm exam number two for math 2270. We're given two matrices A, which is three by three and B, which is three by two, and we're asked to compute the product of A times B. Since we do have a three by three matrix multiplied by three by two matrix, we have a legitimate product here and the product is going to be a three by two matrix when we're done. So to compute A times B, we're going to show all the steps here, one, five, negative three, one, six, negative three, negative one, negative five, four. We just copied down A. We're going to write down B next, two, zero, one, negative one, zero and four. So we're going to take the product of the first row with the first column that when we look at the dot product there, we're going to get one times two, which is two, plus five times one, which is five minus three times zero, which is zero, that's the first entry. Next, we're going to do the first row times the second column. That's going to give us one times zero, which is zero, plus five times negative one, which is negative five, minus three times four, which is negative 12, like so. Next, we're going to do row two times column one, we get one times two, which is two, plus six times one, which is six, and then minus three times zero, which is zero. And then we're going to do row two times column two. We're going to get one times zero, which is zero, plus six times negative one, which is negative six, minus three times four, which is negative 12 again. And then we're going to do the last row. So we're going to take row three times the first column. We get negative one times two, which is negative two, negative five times one, which is negative five, and then four times zero, which is zero. and then lastly we're gonna do row 3 times column 2. We get negative 1 times 0, which is 0, minus 5 times negative 1, which is a positive 5, and then we're gonna get 4 times 4, which is 16. We now want to add these together, all these terms here. So just doing the entries one by one by one. In the 1-1 position, we get 2 plus 5, which is 0, that's a 7. For the 1-2 position, we get 0 minus 5 minus 12, which is a negative 17. Next we're gonna get 2 in the 2-1 position, 2 plus 6 plus 0, which is 8. And then in the 2-2 position, we're gonna get 0 minus 6 minus 12, that's a negative 18. So we move on to the bottom row. In the 3-1 position, we get negative 2 minus 5 plus 0, which is negative 7. And then in the 3-2 position, we get 0 plus 5 plus 16, which is 21. And this is then the product of the 2 matrices.