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# Finding inverse of a 2x2 matrix using determinant & adjugate

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Lets attempt to take the inverse of this two by two matrix, and you'll see that two by two matrices are about the only size of matrices that it's somewhat pleasant to take the inverse of, anything larger than that and it becomes very unpleasant. So the inverse of a two by two matrix is going to be equal to one over the determinant of the matrix, times the adjugate of the matrix, which sounds like a very fancy word but we'll see for a two by two 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 three times two minus negative seven times five, so this is going to be equal to one over three times two minus negative seven times five so minus negative seven times five and then the adjugate of A, and here I'm really just teaching you the mechanics of it, and it's unfortunate that in a typical Algebra II class you kind of have to just go in to 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 two where the three is and the three where the two is so this element right here, this three will go right over there this two, will go right over here and then these two elements, you just take the negative of them so, the negative, let me use a new colour, the negative of, I'm running out of colours, the negative of that is negative five, the negative of that is positive seven so we are left with, this is going to be equal to one over, three times two is six, negative seven times five is negative thrity five, although we have this positive over here, so this whole thing becomes plus thrity five, so six plus thirty five is forty one, so the determinant of our matrix is forty one, we're going to take one over the determinant and multiply it times our adjugate, times two, negative five, seven and three so we get, so this is the drumroll part two over forty one, negative five over forty one I'm just multiplying each of these elements times one over forty one seven over forty one, and three over forty one and we are done!