CHAPTER 05.07: SYSTEM OF EQUATIONS: Rank of a matrix Example 2     In this segment we will look at an example of finding a rank of a matrix. Let’s suppose the problem statement is: What is the rank of this matrix right here? 3, 1, 2, 2, 0, 5, 5, 1, and 7. So again, in this case it is a 3 by 3 matrix. So the largest sub matrix I am going to get from a 3 by 3 matrix, will be 3. That it tells me the rank of A is going to be less than too equal to 3. So the number is going to be 3 or less. Let’s see what the number is. If I look at –Since the largest square sub-matrix I am going to get from here, the order is 3 by 3, that’s A itself-we are going to find the determinate of A. The determinate of A turns out to be 0, in this case. So if I find the determinate of this 3 by 3 matrix, it is 0. So that itself tells me, hey the rank of the matrix is not going to be 3 because that is the only sub-matrix which is a 3 by 3 size and the determinate of that matrix is turning out to be 0. That tells me that, hey, rank of A is maximum of 2, it’s less than or equal to 2.    Let’s go and see if it is 2 or 1,0. What is it? We now have to look at the2 by 2 sub-matrices of this 3 by 3 matrix which is given to us. The first one, which gives us a non-zero we can stop-but if it continues to give us-determined to be 0 then we cannot do so. So let’s suppose if I take the sub-matrix out of this- let’s suppose I take this here, let’s suppose. This is a 2 by 2 matrix. In this case the determinate of 2, 0, 5 and 1 is 2 times one minus 5 times 0, Its 2. That is non-zero. Here I’m finding out the 1st sub-matrix, which I picked up, is of 2 by 2 order, the determinate turns out to be non-zero. So what does that mean? That means the rank of A is 2. I don’t have to look at other sub-matrices which are by the order of 2 by 2 because I am at least able to find one sub matrix which is order by 2 by 2 for the determinate is not equal to 0. And that is the end of this segment.