 Another method of finding the inverse of a matrix is known as the adjoint method. The adjoint method is based on the following theorem. Let A be an invertible N by N matrix. Then the inverse of A is 1 over the determinant times the transpose of some matrix M, where the components of M are the cofactors of the original matrix. Remember, cofactors showed up when we tried to find the determinant. We might say that the inverse is the transpose of the matrix of cofactors scaled by the determinant. For example, consider the 2 by 2 matrix. We'll find the inverse using the adjoint method. For that, we need the determinant, which we found earlier. Now we'll form the matrix of cofactors. Since the cofactors will be the minors of each entry multiplied by plus or minus 1, depending on where in the matrix they're located, it's convenient to set down the pluses and minuses first, and then compute the determinants. So we'll set down our checkerboard pattern of plus and minus, and then we'll find our minors. So the first row, first column entry, we'll cross out that row and column, and the minor will be the determinant of what's left over, which will just be negative 5. The first row, second column entry, we'll cross out that row and column, and the minor will be the determinant of what's left over 4. For the second row, first column will cross out that row and column, and the minor will be the determinant of what's left over negative 3. And for the second row, second column entry, we'll cross out that row and column, and the minor will be what's left over 1. Now we'll incorporate the signs to get the cofactors, and we'll copy down this step, transpose this matrix of cofactors, and at this point we can either write the inverse as 1 over the determinant times the transpose matrix of cofactors, or we can go one further step and multiply the terms of the matrix by 1 over the determinant to get an expression for the inverse matrix. What if we want to find the inverse of a 3 by 3 matrix? As before, the inverse will be the transpose of the matrix of cofactors multiplied by the reciprocal of the determinant. For reasons that'll become apparent, we'll calculate that determinant last. We'll set up our checkerboard of plus and minus, and find our minors. So for the first row, first column, we'll cross out that row and column, and the minor will be the determinant of what's left over. For the first row, second column, we'll cross out the first row, second column to get the minor. First row, third column, we'll cross out the first row, third column, and the determinant of what's left over will be the minor. We'll do the same thing for the entries in the second row and for the entries in the third row. We'll evaluate all of the determinants, take care of the signs to get our cofactors, and then transpose the resulting matrix. Remember, we can think about the inverse as being the transpose of the matrix of cofactors scaled by the determinant. So if we can figure out what that scaling factor is, we'll know what the determinant is. So, notice that if we multiply the original matrix times our transpose of the matrix of cofactors, the first row of the original times the first column of the transpose of the matrix of cofactors is going to give us an entry of 10. And what that means is that the determinant of the original matrix must have been 10 if it's anything at all. And so we'll include that as a factor, which gives us the inverse of the original matrix. It's probably a good idea to verify that this actually is the inverse. But you don't really need to because this is on the internet and it must be true.