00:01
Hello everyone, in this question the matrix a is given as 2 -1, 1, 2.
00:07
We have to construct the matrix exponential or principle fundamental matrix solution.
00:20
So given that x dash equal to ax.
00:23
Therefore, let a equal to 2 -1, 1, 2.
00:29
So we find eigenvalue and eigenvector.
00:32
The determinant of a minus lambda i equal to 0.
00:36
So this implies 2 -lambda, minus 1, 1, 2 -lambda which is equal to 0.
00:43
This implies lambda square minus 4 lambda plus 5 equal to 0.
00:47
This implies lambda equal to 2 plus or minus i.
00:51
So we find eigenvectors.
00:53
So the eigenvectors for two lambdas.
00:59
So the first lambda is 2 -i.
01:03
Then determinant of a minus lambda 1 i equal to 0.
01:09
That is i minus 1, 1, i equal to v1.
01:15
That is eigenvector 1, 0.
01:18
So this implies v1 equal to minus i.
01:25
Now eigenvalue 2 that is 2 plus i.
01:30
Then we have determinant of a minus lambda 2 i equal to 0.
01:36
Therefore, we will be having minus j minus 1, 1, i.
01:47
Minus i minus 1, 1, i...