 In this video, we will provide the solution to question two for exam one for math 2270. We are asked which of the following matrices are in echelon form. So let's first look at matrix A right here. Because this is a non-zero matrix in the first column, we look for the first non-zero column, that's just the first column itself, there would be a pivot in the first column. But because there are non-zero values below the pivot, that would tell you that this matrix is not in echelon form. Let's look at matrix B, similar type of thing. This matrix is non-zero and the leftmost non-zero column is the first column. So there will be a pivot in the first position. But there are non-zero entries below the pivot position. So therefore, this matrix is not in echelon form. If we look at matrix C, this is a non-zero matrix whose leftmost non-zero column is the first column, so we get a pivot right there. There are zeros below the pivot, so so far that's good. The pivot position is one, that's not necessary to be in a row. And you don't need to have a one at the pivot position for echelon form that is required for row-reduced echelon form. If we look at column two, column two, because there's a pivot in the first column and since column two is a non-zero column, then there should be a pivot in the two-two position. That pivot is not a one, that's okay to be in echelon form. There is a zero below it, that's great, so far so good. I'll notice that there's a non-zero entry above it, but in echelon form, that's okay. To be in row-reduced echelon form, we do need to have zeros above the pivots, but that's not required for general echelon form. So then we go to the next column. The next column, there would be a pivot in the three-three position. Notice how our pivots are forming this downward staircase, like so. And so there's a pivot there, it would be, it's just a nine. There's nothing below the nine, so we can't worry about whether zeros are not. So this matrix would be in echelon form. Notice this matrix does not have any rows of zeros. There is the condition that rows of zeros have to be at the bottom. But this matrix is in an echelon form, like there's no rows of zeros. If we look at matrix D, it's a non-zero matrix. The first column is the left most non-zero column. And therefore, there would be a pivot position in the one-one spot. Below the one, there are zeros. The first column is great. Then we move on to the next non-zero column. That would be the second column, which thus puts a pivot position in the two-two spot. There is a zero below the pivot position, so so far so good. We then move to the next column, but there is no zero in this position right here. So we actually don't get a pivot right here. I should say there is a zero there. So and then since there's no other rows we could rotate interchange with, there's no pivot in the third column right here. There is a row of zeros, but you'll notice it's on the bottom of the matrix. Our pivots do make a downward staircase direction, even though there's no pivot in the next column right here. And everything below the pivot is zero. So this is another example of a matrix that is in echelon form. And that would actually indicate to us that the correct answer would be both C and D, which is option F that you can see right here. Now I do want to mention that matrix D is in fact also in row reduced echelon form. Because in addition to being an echelon form, you'll notice that both of the pivot positions are one. And every number above a pivot is also zero. Now, row reduced echelon form is an echelon form. So the fact it's an REF doesn't disqualify it from being part of the answer. It's still an echelon form. Choice C, on the other hand, is not in row reduced echelon form. The reasons for that is that for, of course, there are pivots that are not one. We see that. And we also see that numbers above the pivots are not necessarily zero. So this is an example of matrix in echelon form but not in row reduced echelon form. This matrix is in row reduced echelon form, thus it is in echelon form. So do pay attention to the question. Because the question asked for matrices in echelon form, we would select C and D, which is choice F. If this question instead asked only for matrices in row reduced echelon form, then we would instead choose choice D as the correct answer. So make sure that you do follow the instructions on this one right here.