00:01
The question is to determine the general solution of the system of differential equations x -dash is equal to 1 -3 -negative -2 -negative -4 -x.
00:10
To find the solution of this system, take a to be the matrix 1 -3 -negative -2 -4.
00:17
We have to first find the eigenvalues, lambda -is corresponding to a and the corresponding eigenvectors.
00:24
To determine the eigenvalues solve the equation determinant of a minus lambda i is equal to zero where i is the 2 by 2 identity matrix 1001 so this is determinant of 1 minus lambda 3 negative 2 negative 4 minus lambda is equal to 0 from this obtain the equation 1 minus lambda times negative 4 minus lambda minus 3 times negative 2 is equal to 0.
00:59
This can be simplified to lambda square plus 3 lambda plus 2 is equal to 0, which has solutions.
01:07
Lambda is equal to negative 1 or lambda is equal to negative 2.
01:11
So that the matrix a has eigenvectors, eigenvalues, lambda 1 is equal to negative 1 and lambda 2 is equal to negative 2.
01:20
So that we have to find the eigenvectors corresponding to lambda 1 and lambda 2.
01:26
For that we have to solve the equation.
01:31
A minus lambda i of x is equal to 0 where x not equal to 0 is the eigenvector corresponding to lambda.
01:43
Now consider lambda 1 is equal to negative 1...