00:01
In this question, you're asked to find the basis for the eigenspace of the matrix a corresponding to the eigenvalue lambda equals to 5.
00:13
Recall that an eigenvector is a vector satisfying the equation a x equals lambda x.
00:21
Here, x is a non -zero solution and lambda is some fixed number.
00:28
So in our case, lambda equals to 5.
00:30
We want to solve the equation a x equals 5x.
00:36
And we can rewrite this equation as a x minus 5x equals 0.
00:41
And after factoring out x, we are going to get a minus 5i, where i is the identity matrix times x equals 0.
00:50
So this is a system of equations we want to solve.
00:54
But first we need to calculate a minus 5i.
01:01
To do that, we need simply to subtract 5 from the entries on the main diagonal of the matrix 8.
01:08
So we will get 1, negative 4, 15 for the first row, 2, negative 4, 14 for the second row, and 2, negative 4, 14 for the last row.
01:38
And the right -hand side is going to be 0.
01:40
So this is the matrix a -1 -5 -i...