 In this video, we're going to provide the solution to question number five from exam three for math 2270 Which of the following matrices has real eigenvalues? There are three matrices provided rs and t we have to decide which of them has real eigenvalues And which ones do not there are a few things we can do to find eigenvalues very quickly because after all worst-case scenario if we have a Matrix that's two by two matrix We could always just compute the characteristic polynomial and find its roots, which are the eigenvalues That's sort of like a worst-case scenario this question's intending for us to be able to discover Eigenvalues a little bit quicker than that so what are some some of the tricks of the trade here when you look at when you look at r? You'll notice that r is an upper triangular matrix because you have a zero below the diagonal right there If you have an upper triangular matrix a lower triangular matrix or a diagonal matrix Which is both upper and lower triangular the eigenvalues are going to be those numbers along the diagonal only for triangular matrices here And so we see that the eigenvalues For r are going to be the numbers one and three in particular Those are real numbers and so we see that r will be a matrix with real numbers So we should include that in our solution when we look at Matrix s you have one two two one This is an example of a symmetric matrix for which we've learned that symmetric matrices always have real eigenvalues So that's one of the easiest things to detect. I don't even have to compute the characteristic polynomial of the eigenvalues I know they'll be real even though I have no idea what they are at the moment Whenever you find a symmetric matrix, it'll have real eigenvalues That's also true if you have a Hermitian matrix for which a Hermitian matrix is gonna be a complex matrix Which is equal to its own conjugate transpose So I don't want you to assume just because you see a complex matrix with non real entries that it means the eigenvalues are non real No, no, no, no Hermitian matrix is exactly a Non real matrix whose eigenvalues can be real. So so far we have r and s and if we look at The list of possibilities here, you'll notice that there is no solution that says r s and t So we can already see at this moment that the correct answer would have to be d R and s by process of elimination, but why does t not work? Well t is an example of a rotation matrix Just as a reminder. This is a rotational matrix a matrix of the form cosine theta negative sine theta sine theta and cosine theta now I sometimes forget Where does the negative sign go does it go here or here, right? And if you move the negative sign from from the top right position to the bottom left position You're just gonna switch it from a counterclockwise rotation to a clockwise rotation So it still would be a rotation matrix just yet the patient to the direction where this is the standard matrix for rotation in the plane about the origin by the Angle theta this right here is an example of a 90 degree rotation counterclockwise right here Why is this relevant to our discussion of real eigenvalues here? Well, one thing to report is that a rotation matrix never has real eigenvalues The only exception of that would be zero degrees and a hundred and eighty degrees And that's because if you have a zero degree rotational matrix that's just the identity matrix and that situation with the eigenvalues are one and one and If you'd rotate a hundred eighty degrees, that's actually just equal to negative One times the identity matrix in which case your eigenvalues are negative one negative one both Both of these are example of diagonal matrices for which we would invoke the previous principle But for all these other Rotational matrices we get that they don't have real eigenvalues There's no vector when you rotate it which will be a scalar multiple of its original using just real numbers In fact the eigenvalues of this 90 degree rotation are going to be plus or minus i which are not real numbers You can find that out again directly from the Characteristic polynomial, but again this question is intended for us to be able to find The eigenvalues whether they're real or non real without having to calculate the characteristic polynomial So rotation matrices. Nope not going to help us out here We can see the eigenvalues from a triangular matrix and we know the eigenvalues of a symmetric or Hermitian matrix are always real So D would be the correct response