00:01
Alright, so this question covered topic relating to matrix exponential and the eigenvalues, eigenvectors, and a system of differential equations.
00:11
From here you have a system of differential equation.
00:14
First, you need to rewrite it as x1, x2, fine, equal to the matrix a in this case is equal to 12, negative 12, negative 6, x1, x2.
00:29
Notice that i do know xt as a function, the vector function, with two coordinates x1 and x2, like this.
00:39
Okay, and this is our matrix a.
00:44
Right, so we are going to find exponential of at, which means that we have to find.
00:51
First, you need to verify whether a is diagonalizable, right? what does it mean by an antagonizable? meaning that it has two different eigenvectors.
01:01
In this case, when you compute the determinant of a minus lambda i equals 0, you will get two solutions, lambda equal to 0 or lambda equal to 6.
01:13
So lambda is zero and equal zero is an eigenvalue.
01:17
Lambda equal to 6 is another eigenvalue, right? so this metric a is diagonalizable, which means that a can be rewritten as p, a, p minus 1...