### Video Transcript

Calculate the eigenvalues of the
matrix ๐ด equals negative one, zero, zero, negative one.

Well, first when we take a look at
our matrix, we can see that itโs a two-by-two matrix. Well, we know that an ๐ by ๐
matrix will always have ๐ eigenvalues. So therefore, we know that with our
matrix being a two-by-two matrix, weโre gonna have two eigenvalues. Okay, great, so we know thatโs what
weโre looking for. But what can we do to work out the
eigenvalues? So to help us solve this problem
and find our eigenvalues, we have a relationship. And that one is that the
determinant of the matrix ๐ด minus ๐๐ผ, where ๐ผ is the identity matrix โ โ well,
this is equal to zero. And Itโs the ๐ in our relationship
thatโs gonna help us calculate our eigenvalue.

So therefore, using our
relationship, we can say that the determinant of the matrix ๐ด, which is negative
one, zero, zero, negative one, minus ๐ multiplied by the identity matrix, which is
the matrix one, zero, zero, one, is just gonna be equal to zero. And as we said, the identity matrix
is one thatโs in this form. But we have these terms as one. So the diagonal running from the
top left to bottom right is one. So therefore, if we multiplied by
the scaler, which was our ๐ , weโre gonna get negative one, zero, zero, negative
one minus ๐, zero, zero, ๐. And we want to find the determinant
of this. So once we carried out the
subtraction of our matrices, weโre gonna get the determinant of the matrix. Negative one minus ๐, zero, zero,
negative one minus ๐ is equal to zero.

So when we look at a two-by-two
matrix and we want to find the determinant of that, if you remind ourselves what we
do, well, we multiply diagonally and then subtract. So we have ๐๐ minus ๐๐. So if we have the matrix ๐, ๐,
๐, ๐ the determinant of that is ๐๐ minus ๐๐. So therefore, if we do that, weโre
gonna get negative one minus ๐ multiplied by negative one minus ๐ minus zero
equals zero. So then, if we distribute the first
parentheses over the second parentheses, weโre gonna get โ well, first of all, weโre
gonna get one plus ๐. Thatโs cause negative one
multiplied by negative one is one and negative one multiplied by negative ๐ is just
๐. So itโs one plus ๐. Then weโve got plus another ๐ and
then plus ๐ squared. Then this is equal to zero. So then if we tidy this up, we get
๐ squared plus two ๐ plus one equals zero.

So now what we need to do is solve
to find ๐ cause itโs gonna give us our eigenvalues. Well now to solve this, what we can
do is factor because we can factor our expression because weโve got ๐ squared plus
two ๐ plus one. Well, we need two values. Theyโre gonna multiply together to
give us one and add together to give us two. So therefore, weโre gonna get ๐
plus one multiplied by ๐ plus one is equal to zero. And thatโs because if we have one
multiplied by one is one; one add one is two. So they satisfy this. So now what we need to do is solve
this to find our ๐ value. Well, what weโre gonna do is set
one of our parentheses equal to zero. And thatโs because to get a result
of zero, we need to have zero multiplied by something to get zero. So Iโm gonna have ๐ plus one
equals zero.

So therefore, ๐ is gonna be equal
to negative one. And actually, as you can see in
this one, weโve got ๐ plus one multiplied by ๐ plus one. This is same as ๐ plus one
squared. So weโve got repeated roots. However, we were expecting two
eigenvalues cause we showed that earlier. There should be two eigenvalues for
two-by-two matrix. But weโve got a repeated
eigenvalue, so repeated root. So therefore, what we say is that
our eigenvalue, or eigenvalues, is negative one. But negative one is degenerate. And itโs degenerate because both of
our eigenvalues are identical because weโve got, as we said, repeated roots.