Finding inverses of 2x2 matrices

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 adjective the matrix which sounds like a very fancy word but we'll see for about 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 3 times 2 minus minus negative 7 times 5 so minus negative 7 times 5 and then the adjutant a and here I'm really just teaching you the mechanics of it and it's a little unfortunate that in a typical algebra 2 class you kind of have to just go into the mechanics of it but at least it'll this will get us to where we need to go so the adjutant need to swap the two elements on this diagonal so put the 2 where the 3 is and 3 where the 2 is so let me let me so this element right over here this 3 will go right over there this this 2 will go right over here and then these 2 elements you just take the negative of them so the negative let me do a new color the negative of 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 five so six plus 35 is 41 so the determinant of our matrix is 41 we're going to take one over the determinant and multiply it times our ad you get times 2 negative 5 7 and 3 so we get so this is the drumroll 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 or 41 and we are done