CHAPTER 05.06: SYSTEM OF EQUATIONS: Rank of a matrix Example 1     In this segment we are take an example how to find the rank of a matrix. So let’s suppose the problem statement is – hey - what is the rank of this particular matrix here 3, 1, 2, 2, 0, 5, 1, 2, 3. So right here you can see that it’s a three by three matrix so the largest sub-matrix which I’m going to order of the sub-matrix, which I’m going to get by a three by three matrix will be a three by three so that itself tells me that rank of (A) is going to be less than or equal to 3. So it can be a maximum of 3 but it might be less than that; so how do we find out what is the rank of (A)? If I look at the determinant of (A), because the three by three matrix, and that’s the largest sub-matrix which I can find out of that the determinant of (A) itself is minus 23 and you can always figure out how to find the determinant of a matrix. So it’s minus 23; so not equal to zero; so in this case the rank of the matrix the rank of (A) is just 3. And that’s the end of this segment.