 So, one of the things we have to be able to do is to be able to find matrix inverses. And there's a couple of ways of doing that, but probably the fastest in certain circumstances is what's called the cofactor method. And so that works as follows. So here I have this matrix 3, 5, 2, 1, and I want to find the inverse of the matrix. So I can find that inverse using the following steps. First, I'm going to form the matrix of minors, and we'll explain what that is. I'm going to form then the matrix of cofactors, and then I'm going to transpose this last matrix, and I'm going to scalar multiply by the reciprocal of the determinant of the original matrix. So let's go through those steps. So this matrix of minors, that's going to be the determinant of the minor of each entry. So how do I form that minor? Well, whatever the entry is, what I'm going to do is I'm going to wipe out the row and column. And what's left over is the minor, and this matrix of minors is going to be the determinant of what's left over. So we'll go ahead and find that. So our first entry here, this 3, I'm going to wipe out the row and column that it's in. What's left over is this, and the determinant of a 1 by 1 matrix is just the value of the matrix itself. Our next entry, 5, I'm going to wipe out the row and column that it's in, and my left over matrix is 2 with determinant 2. Again, I'm going to take this entry, obliterate the row and column, 5, determinant is going to be 5. And then finally, that last entry, wipe out row, wipe out column. I have this, and the determinant is going to be 3. So there's my matrix of minors. Now I'm going to form the matrix of cofactors. And to do this, what I'm going to do is I'm going to assign plus and minus in a checkerboard pattern to produce this matrix of cofactors. So I'll start in the upper left-hand corner with a plus and then alternate signs all the way through. And there's our matrix of cofactors. Now the next step, I'm going to just transpose that matrix, so I'm going to flip it. Now note that the signs now get attached to their numbers. So this is negative 5, negative 2, positive 3. I'm not going to write the positives. But I'm going to flip along the diagonal, and there's my transpose of the matrix of cofactors. And so my last step is going to be multiplying by the reciprocal of the determinant. So my determinant of the original matrix 3521 is going to be the difference of the products of the diagonals, 3 times 1 minus 5 times 2, negative 7. So I'll multiply everything by negative 1 over 7, and that will give me the reciprocal, the inverse of this original matrix. There's my minus 1, 7 times this transposed matrix of cofactors. And I can move the negative 1, 7 factor inside to give us a proper looking matrix as our inverse. Nothing really changes if we go up to a 3 by 3 matrix. The only thing that happens is that things get a little bit more complicated. So there's my 3 by 3 matrix that I want to inverse. So I'll start by forming the matrix of cofactors. So again, I'll start with each entry, start with this negative 2. I wipe out the row and column, leaving a minor, and that is going to be this. I'll find the determinant, and the upper left entry starts with a plus. So my first entry there is plus determinant of the minor. What's the next entry going to be? Well, there's my next entry, negative 1 here. Wipe out column and row, checkerboard pattern plus, minus. And what I have left when I wipe out row and column is this matrix 0, 4, negative 3, 1. So I want negative the determinant of that minor. And I'll continue in this fashion. The next entry is going to be a plus, and wipe out row and column gives me this matrix 0, 4, 1, 2, and I want to find the determinant of that. Continuing, wipe out row and column, wipe out row and column, wipe out row and column, and so on. And this gives me my matrix of cofactors, and I just need to evaluate what those determinants are. So evaluating the determinants of each of those matrices gives me this as my matrix of cofactors. I'll transpose it and multiply by the reciprocal of the determinant of the original matrix to get the inverse. And that reciprocal of the determinant is 1 over negative 22. And so there's my inverse of the matrix.