Welcome to an example
on how to find a trace and determinant
of a three by three matrix.
We'll find the trace and determinant
using the more formal definition
of a trace and determinant.
Then we'll also compare how we can find
the trace and determinant
using the eigenvalues
of the given square matrix.
The eigenvalues for
the given square matrix
are two and four,
though the eigenvalue of two
has a multiplicity of two.
We've already found several eigenvalues
for three by three matrices,
but if you do want to see the work
on how to find these eigenvalues,
I've show it here.
In general, to find the
eigenvalues of a square matrix,
we need to solve the equation shown here,
which is a determinant of the difference
of lamba I and A equals zero.
Again, here's the work that shows
the eigenvalues are two and four,
where the eigenvalue of two
has a multiplicity of two.
So one way to find the
trace of a square matrix
is to find the sum of the elements
along the diagonal given
by A sub one comma one,
A sub two comma two, all the
way through A sub n comma n.
In our case, because we have
a three by three matrix,
the trace of matrix A would be the sum
of these three elements here.
So using this definition of the trace,
we can say the trace of matrix A
is equal to two plus three plus three
which equals eight.
But the trace of a square matrix
is also equal to the
sum of the eigenvalues.
And because we know the
eigenvalues of a given matrix,
we could also find the trace,
by summing the eigenvalues,
which will be two plus two plus four,
which of course is also equal to eight.
Now let's find the determinant
of the given three by three matrix.
We've already found the determinant
of several three by three matrices
using the diagonal method,
as well as the cofactor
method explained here.
So to show how to find the determinant
using the cofactor method,
look I've already set this
up and shown the work here,
so you may want to pause
the video and review this.
The determinant of the square
matrix A is equal to 16.
But another way to find the determinant
using eigenvalues is shown here,
where the determinant of a square matrix
is equal to the product
of the eigenvalues.
In this case, because we have
the three by three matrix
we can find the determinant
by finding the product
of lambda sub one, lambda sub
two, and lambda sub three.
So if we have the eigenvalues,
another way to find the
determinant for matrix A
would be to find the product
of two, two, and four.
Well two times two times four is also 16,
the value of the determinant.
I hope you found this helpful.
