00:01
In this question, we have to find the eigenvalue and vector for the matrix 2, 1, 1, 0, 4, 1 and 0, 0, 6.
00:11
6, okay, so first of all, we consider determinant of a minus lambda i equals to 0, this is determinant of 2 minus lambda 1, 1, 0, 4 minus lambda 1, then we have 0, 0, 6 minus, since this is upper triangular, determinant is just equals to the product of diagonal element, so we have 2 minus lambda 4 minus lambda and 6 minus lambda equals to 0 we get lambda value as 2 4 and 6 these are the required eigenvalues next we have to find the eigenvector.
00:52
So the eigenvector for here we find lambda equals to 2 we consider ax equals to 2 times of x this will implies a minus 2 2i x equals to 0.
01:08
So what we are going to get here, this will be 2 minus 2 is 0, 1, 1, 0, 4 minus 2 is 2, 1, 0, 0, 6 minus 2 is 4 times, let's consider this eigenvector as x1, x2, x3 equals to 0 from this we get 4 times of x3 equals to 0 then we have 2x2 plus x3 equals to 0 and x2 plus x3 equals to 0 from these three we get that x2 and x3 they both are 0 and x1 we can take at parameter t okay so let t equals to 1 this implies that eigenvector will be 1 0 0 now let us find out the eigenvector for lambda equals to 4 we have here a minus 4 times i x equals to 0 so we have 2 minus 4 is minus 2 1 1 0 0 1 then we have 0 0 6 minus 4 is 2 times x1 x2 x3 equals to the zero matrix again if we see we have x3 equals to 0 then we get 2 times x3 sorry equals to 0 and we have minus 2x1 plus x2 plus x3 equals to so this implies 2x1 plus x2 equals to 0.
02:53
So we consider x2 equals to, let's say it is equals to t.
03:01
Then we have x1 value equals to t over 2.
03:08
Let us take t value as 2.
03:11
We will get x2 as 2 and x1 as 1.
03:14
So we have the eigenvector.
03:15
So this was the eigenvector for lambda equals to 2 and here we get 1 x2 is 2 x3 is 0 this is one of the eigenvector for lambda equals to 4 similar way we have to find for lambda equals to 6 so let us see that also for lambda equals to 6 we we consider a minus 6i times x equals to 0.
03:45
So a minus 6i, here we get minus 411, 0 minus 2, 1, 0, 0, 0 times x1, x2, x3 equals to the 0 matrix...
