00:01
The first thing we're going to do here is find the eigenvalues of our matrix a.
00:04
And to do that, we're going to look at the determinant of our matrix a minus our eigenvalues lambda times the identity matrix, which will be the determinant of 2 minus lambda, 0, 0, 0, 5 minus lambda, negative 7, 0, 0, 2, negative 4 minus lambda for our matrix.
00:32
And since this is a 3x3 matrix, we can almost choose how we calculate it.
00:38
And to do so, i'm going to pick this term to start with.
00:43
And if you do so, you have to multiply that term by these four, the determinant of the bottom right of this matrix.
00:53
And this will be 2 minus lambda times the determinant of the bottom 4, which will be 5 minus lambda times negative 4 minus lambda.
01:06
Minus the off -diagonal terms, which will give us positive 14, minus the term next to the term we started with.
01:17
So 0 times the determinant of these four points, but 0 times anything is just 0.
01:24
So this will just be 0, plus the term to the right of that one, times the determinant of these four points, which will also just be 0.
01:35
So plus 0.
01:36
So here we only needed to do one calculation.
01:38
So now we want to know when 2 minus lambda times 5 minus lambda times negative 4 minus lambda plus 14 is equal to 0.
01:54
And we can notice that we have a 2 minus lambda times a bunch of other stuff.
02:01
So we know that if lambda is 2, we have this whole equals a 0.
02:06
So 2 is an eigenvalue.
02:08
And now we can look at when the inside term here is zero.
02:13
And foiling this out will give us negative 20, minus 5 lambda, plus 4 lambda, plus lambda squared, plus 14.
02:28
We want to know when this is equal to zero.
02:31
If we combine terms, we're going to have lambda squared minus lambda, minus 6 is equal to zero.
02:38
And we can factor this on the left to be lambda minus 3 and lambda plus 2.
02:46
And then, therefore, we have two more eigenvalues when lambda is 3 and when lambda is negative 2.
02:56
So now we have to go through and find the three eigenvectors that could correspond to these eigenvalues.
03:01
So we're going to start with lambda 1, which is equal to 2.
03:04
So we're going to solve for the eigenvector by plugging in 2 to the matrix we took the detour.
03:11
Of up here.
03:13
So this will be 2 minus lambda, which is 2, 0, 0, 0, 5 minus 2, and negative 7, and then 2, and then negative 4 minus 2 times our eigenvector v1.
03:32
And we want to know when this is equal to 0...