Finding inverse of a 2x2 matrix using determinant & adjugate

Let's attempt to take the inverse of this 2 by 2 matrix. And you'll see the 2 by 2 matrices are about the only size of matrices that it's somewhat pleasant to take the inverse of. Anything larger than that, it becomes very unpleasant. So the inverse of a 2 by 2 matrix is going to be equal to 1 over the determinant of the matrix times the adjugate of the matrix, which sounds like a very fancy word. But we'll see for by a 2 by 2 matrix, it's not too involved. So first let's think about what the determinant of this matrix is. Well, we've seen this before. We just look along the two diagonals. It's 3 times 2 minus negative 7 times 5. So this is going to be equal to 1 over 3 times 2 minus negative 7 times 5. And then the adjugate of A-- and here I'm really just teaching you the mechanics of it. And it's a little unfortunate that in a typical algebra II class you kind of just go into the mechanics of it. But at least this will get us to where we need to go. So the adjugate of A, you literally just need to swap the two elements on this diagonal. So put the 2 where the 3 is and the 3 where the 2 is. So this element right over here, this 3 will go right over there. This 2 will go right over here. And then these two elements, you just take the negative of them. So the negative-- let me do a new color. Actually, I'm running out of colors. The negative of that is negative 5. The negative of that is positive 7. So we are left with-- this is going to be equal to 1 over-- 3 times 2 is 6. Negative 7 times 5 is negative 35. But then we have this positive over here. So this whole thing becomes plus 35. So 6 plus 35 is 41. So the determinant of our matrix is 41. We're going to take 1 over the determinant and multiply it times our adjugate, times 2, negative 5, 7, and 3. So we get -- so this is the drum roll part -- 2 over 41, negative 5 over 41. I'm just multiplying each of these elements times 1 over 41. 7 over 41, and 3 over 41. And we are done.