Answer :
The solution for the linear system will be , x(t) = 4 c₁e^t + 4 c₂e^t and
y(t) = -c₂e^t + c₂(1 - t)e^t .
The solution is given by ,
matrix | x y | = c₁ .v . e^λt + c₂( w + v.t ) e^λt
Given that the matrix has repeated eigenvalue with eigenvector generalized vectors ,
λ = 1 with eigenvector v = [ 4 , -1 ] and generalized vectors w =[ 0,1 ].
then the solution will be,
c₁ [ 4 , 1 ] e^t + c₂[ 0 , 1] + [ 4 , -1 ] )e^t
therefore,
x(t) = 4 c₁e^t + 4 c₂e^t
y(t) = -c₂e^t + c₂(1 - t)e^t
Matrixes represent linear maps and allow for explicit linear algebra operations. As a result, matrices play an important role in linear algebra, and most characteristics and operations in abstract linear algebra may be represented in terms of matrices.
Matrix multiplication, for example, depicts the combination of linear maps.
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