## Matrix Differential Equation

1. The problem statement, all variables and given/known data
The questions are in the image

3. The attempt at a solution
My solutions are
V1=3*(1 -2)e-2t+ (-2) (1 -3)e-3t
V2=1*(1 -2)e-2t+ (-1) (1 -3)e-3t

How do I get the X matrix since my solutions are in exponential still.

Thank you for all the help
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 According to your solution, $$X=(v_1\ v_2)=\begin{pmatrix}3e^{-2t}-2e^{-3t}&-6e^{-2t}+6e^{-3t}\\ e^{-2t}-e^{-3t}&-2e^{-2t}+3e^{-3t}\end{pmatrix}.$$

 Quote by Some Pig According to your solution, $$X=(v_1\ v_2)=\begin{pmatrix}3e^{-2t}-2e^{-3t}&-6e^{-2t}+6e^{-3t}\\ e^{-2t}-e^{-3t}&-2e^{-2t}+3e^{-3t}\end{pmatrix}.$$
That means I just plug in t=0 to prove that x(0)=(1 0
0 1) and also use the same method to prove that dx/dt =AX

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## Matrix Differential Equation

You have the columns and rows swapped.

 Quote by vela You have the columns and rows swapped.
Even after multiplying it I get this respective solutions, what do I do next?
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 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus You can write v1 as a single vector: $$\vec{v}_1 = \begin{pmatrix} 3e^{-2t} - 2e^{-3t} \\ -6e^{-2t} + 6e^{-3t}\end{pmatrix}$$Do the same for ##\vec{v}_2##.

 Quote by vela You can write v1 as a single vector: $$\vec{v}_1 = \begin{pmatrix} 3e^{-2t} - 2e^{-3t} \\ -6e^{-2t} + 6e^{-3t}\end{pmatrix}$$Do the same for ##\vec{v}_2##.
i get that part but after that what do I do to get just numbers in my 2x2 matrix so that I can prove dx/dt =AX and x(0)= \begin{pmatrix}1 0\\ 0 1\end{pmatrix}

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Like the problem says, the first column of X is v1. Its second column is v2. It's not going to be just numbers. I'm not sure why you think it has to be.

 Quote by vela Like the problem says, the first column of X is v1. Its second column is v2. It's not going to be just numbers. I'm not sure why you think it has to be.
Okay but then if I put t=0 into the x equation, I do not get the identity matrix and how would I verify that dx/dt=AX by just differentiating the X matrix?.

Thank you for all the help

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Show us how you're calculating X when t=0.

 Quote by vela Show us how you're calculating X when t=0.
Okay I got just did a arithmetic error, but how do I verify that dx/dt=AX

 Recognitions: Gold Member Homework Help Science Advisor Staff Emeritus Calculate both sides and show they're equal to each other.
 Okay I will give it a try, thank you very much for all the help.