 In this video, we will present the solution to question number 13 from the practice exam number three from math 2270 We're given a matrix a which in this case is two by two. It's negative five eight negative four seven We are given eigen pairs for a that is this is an eigen value with a corresponding eigen vector Here's another eigen value with its corresponding eigen vector So we see that a times the vector two one gives you negative one times two one and a times the vector one one Gives you three times one one. So we have these eigen pairs And we're asked to compute a diagonalization of a which means we need to find the factorization of a where a equals PDP inverse Where P is a non-singular matrix so that P inverse makes sense and D is a diagonal matrix So how would you how do we compute this? diagonalization well the things to remember here is that D is gonna be a diagonal matrix, right? It's size will be the same as a so it'll be a two by two diagonal matrix The numbers along the diagonal are gonna be the eigen values of a which we see the eigen values are negative one and three So the diagonal matrix D will be negative one and three you can switch the order if you want to but it doesn't matter Just just just put the diagonals along the put the eigen values on the diagonal I'll just take the same order that they were presented to us Then the matrix P Is going to coincide with an eigen basis of the corresponding eigen values So since we know that two one is an eigen value eigen vector for the eigen value negative one The first column of P is gonna be two and one the second column I'm gonna take to be one one because one one is the eigen vector associated to the eigen value three Now like I said, it didn't matter which order you did D here You could do you could you could do whatever order you want just make sure that these These column vectors are in the same order So that you always have an eigen pair So the first column should make an eigen pair the second column should make an eigen pair If we have a three by three matrix then the third column should make an eigen pair go on from there Now to find P inverse This is honestly the hardest part of the problem here to find P inverse. We need to compute the inverse of P for which you can do the Inversion algorithm for which if we take P augment the identity here You're gonna row reduce that you would get the identity Augment P inverse so you can row reduce that to to find the P inverse that might be necessary If this matrix is kind of big like three by three four by four or bigger as this is a two by two matrix I'm gonna use the formula that P inverse is Gonna equal one over the determinant of P Right which the term in a P. I'm gonna take that to be you're gonna take the diagonals minus the diagonals there So you get whoops Two minus one so the determinant is going to turn out to be one and then you get the adji kid right here for which Remember you're gonna swap the locations of the diagonal entries So you get a one and a two and then the off diagonal entries You're just gonna negate them so you get negative one negative one and so since the determinant turned out to be negative one We see that P inverse is gonna be one negative one negative one negative one and two for which It I would encourage you to double check your answer if you can PP inverse right you can always check it at the very least put into your calculator, right? One negative one negative one and two right here, which notice if you take the first row first column They're gonna get two minus one which is one first row times the second column you get negative two plus two Which is zero second row first column you go one minus one, which is zero Second row second column you get negative one plus two which is one so that is in fact the identity Like so when you're doing this question, you can't simply just throw in the matrix into your graphing calculator to compute the inverse matrix I do need to see the steps of computing the inverse matrix for full credit Then we see that the matrix a will equal PDP inverse Give them as the factorization write it out. We get two one one one We're gonna get so we get P first in the diagonal negative one zero And three and then we get P inverse which turned out to be one negative one negative one and two like so And so here gives us the diagonalization of the matrix This was a two by two case, but doing three by three four by four would not be much more difficult Because on this question you will definitely be given all of the eigen pairs you need You'll be given an eigen basis with their corresponding eigen values from which case then you build the diagonalization computing the inverse of P is gonna be the hardest step there You