Inverse of 3x3 Matrix transcript

okay so we're going to find the inverse of a this is the same Matrix as the gor elimination now we're going to use the other technique a long it's a bit longer but it gets the same result okay so first we need to think of the inverse of a is the adjoint of a over the determinant of a so the adjug a is the transpose of the co-actor Matrix we will get to that shortly let's work out what the determinant of a is first so we're going to say uh determinant of a is there's no real shortcut here let's just expand Row one two and the leftover Matrix is 21 one 2 21 1 2 minus 1 31 1 2 31 31 2 two very to make a mistake one 3 2 two 1 okay 2 a - BC 2 2's are 4 - 1 -1 2 3 is are 6 - 2 3 1's are 3 - 4 2 * 3 - 1 * 4 + 1 * 6 - 4 - = one so we can now say that the determinant is one okay a joint a okay so the first thing we need to find is the co-actor Matrix co-actor Matrix let's get us some room here okay so instead of writing the two we're going to cover up Row one and column one which the two is in and that's our leftover numbers and that goes into the determinant 2 one one two and next one we're going to cross out where the one is and the left over is 31 2 two and so forth 3 two two 1 and we're going to keep going without making a mistake easier said than done you'll notice I'm not concerned about where the negatives are just yet and Row one column one one + 1 is 2 even so that's positive Row 1 column 2 1 + 2 is three odd so that's negative uh Row one column 3 1 + 3 is 4 that's even so that's positive row two column 1 2 and one is three you can see the pattern me okay so our co-actor Matrix is becomes a minus BC so that's going to be 4 -1 the negative of 6 - and 3 - 4 the negative of 2 -1 4 - 2 the 2 - 2 1 - 2 negative of 2 - 3 and 2 2 4 - three and get some room here three - 4 -1 -1 2 -1 1 - 4 sorry 3 - 4 -1 -1 2 0 and -11 1 becomes our co-actor Matrix but we're wanting a joint a so we've got to take that co-actor Matrix and we need to transpose it so remember a joint a is the transpose of the co-actor Matrix so we leave our leading diagonal exactly the same 3 2 1 this1 and -4 change places the two negative ones change places and the one in the zero change place that becomes our transpose of our co-actor Matrix which is our a joint [Applause] a now now that we have a joint a over the determinant remember determinants one so that becomes our inverse Matrix I Che it a little bit because when we have divide by one we don't change so that becomes our inverse Matrix and hopefully that agrees with what we got with Gan elimination I'm sure it does um and that's the process the long-and process of getting the inverse Matrix please not that um you're not always going to get a a determinant of one so guess you're going to have some fractions in there but it's all good fun isn't it