00:01
We are given with a system of linear equation that is y prime is equals to a matrix.
00:11
Matrix that is has entries minus 7, minus 5, 9, 5 with vector y plus another vector that is e power minus t minus 4 e power minus t.
00:36
We have to calculate the fundamental matrix for this.
00:41
For fundamental matrix we must know the eigenvalues of this matrix.
00:46
For calculating eigenvalues we have to calculate a minus x time identity matrix it's a determinant equals to 0.
00:54
If we put the determinant equals to 0 we get equation that is x square plus 2x plus 10 is equals to 0.
01:05
From here we get x value is minus 1 plus 3 iota and another is minus 1 minus 3 iota.
01:14
Now we will calculate eigenvectors corresponding to these eigenvalues.
01:19
For that a minus lambda 1 eigenvalue into identity matrix multiplied with vector v is equals to 0.
01:29
If we calculate we get our matrix will be minus 6 minus 3 iota minus 5 this as it is 9 as it is and this is 6 minus 3 iota multiplied with vector.
01:47
Vector can be x and y make it equals to 0.
01:52
If we put equals to 0 we get equations are minus 6x minus 3 iota x minus 5y is equals to 0 and second equation will be 9x plus 6y minus 3 iota y is equals to 0.
02:13
From first equation we will get y will be equals to minus 6x minus 3 iota x by 5...