We're going to find the eigenvalues
of the given two-by-two matrix.
If we let A be an n by n matrix,
then the number lambda
is an eigenvalue of A
if and only if the determinant
of lambda times the identity
matrix minus A equals zero,
or equivalently, the determinant
of A minus lambda I equals zero.
Let's go ahead and use this equation here
to determine the eigenvalues.
So we'll have the determinant
of lambda times I,
because matrix A is a two-by-two matrix
would be lambda times the
two-by-two identity matrix.
And therefore this
matrix is lambda times I.
So the first row is lambda zero
and the second row is zero lambda.
Then we have minus the given matrix A.
So to find the eigenvalues,
this determinant must equal zero.
Let's go ahead and write the determinant
using vertical bars,
where the first row would be
lambda minus negative
22 or lambda plus 22.
And then we have zero minus three,
so here we have negative three.
The second row we have
zero minus negative 123.
That's positive 123.
The last element is lambda minus two.
So the value of this
two-by-two determinant
is this product minus this product.
So our equation, called
the characteristic equation
is the quantity lambda plus 22
times the quantity lambda minus 2
minus negative three
times 123 equals zero.
Let's go ahead and multiply
these two binomials.
So we have lambda squared minus two lambda
plus 22 lambda minus 44.
And then here we'll
have minus negative 369
or plus 369 equals zero.
So combine like terms.
We have lambda squared plus 20 lambda
plus 325 equals zero.
Let's continue solving this
equation on the next slide.
Because there are no factors
of 325 that add to 20,
this is not factorable,
so we'll have to use the quadratic formula
in order to find these solutions.
Our a equals one,
b equals 20 and c equals 325.
So of course instead of
x, we'll have lambda.
So we'll have lambda equals...
Negative b would be negative 20
plus or minus the square root
of b squared, that'd be 20 squared,
minus four times a, which
is one, times c which is 325
all divided by two times a,
which would be two times one.
So simplifying, here we have negative 20
plus or minus the square root
of this is going to be 400
minus 1300
all divided by two.
So lambda is equal to negative 20
plus or minus the square root of
this is going to be negative
900, divided by two.
The square root of negative 900
is equal to the square root of 900
times the square root of
negative one, which equals 30 i.
So this simplifies to negative 20
plus or minus 30 i divided by two.
Breaking this up into the real
part and the imaginary part,
we have lambda equals
negative 20 divided by two
plus or minus 30 i divided by two,
which equals negative
10 plus or minus 15 i.
So we have two eigenvalues.
Lambda one is equal to
negative 10 plus 15 i,
and lambda two is equal
to negative 10 minus 15 i.
But if we go back to
the original question,
I believe we're only
supposed to enter a and b.
And so we're told here the
eigenvalues are complex numbers.
In our case, a is equal to
negative 10 and b is equal to 15.
Notice how we don't enter
the plus or minus or the i.
We're only entering the value
of a and the value of b.
I hope you found this helpful.
