How to Solve System of Equations with Matrix Y

  • MHB
  • Thread starter karush
  • Start date
  • Tags
    System
Thank you!In summary, to solve the given system, we use the method of diagonalization by finding the eigenvalues and eigenvectors of the matrix A. Then, we construct the diagonal matrix D and the matrix of eigenvectors P. Finally, we use the formula Y = P*D*P^-1 to find the solution to the system Y(t) = e^(At)Y(0).
  • #1
karush
Gold Member
MHB
3,269
5
nmh{383}
307.27.3 Solve the system
$$Y'=\begin{bmatrix}
1 & 3 & -3 \\
0 & 1 & 0 \\
6 & 3 & -8
\end{bmatrix}Y$$
ok how do we get Y
 
Last edited:
Physics news on Phys.org
  • #2


Hello,

To solve this system, we will use the method of diagonalization. First, we need to find the eigenvalues and eigenvectors of the matrix A = [[1,3,-3],[0,1,0],[6,3,-8]].

The characteristic equation of A is det(A-λI) = 0, where I is the identity matrix. Solving this equation, we get the eigenvalues λ = 1, 1, -2.

Next, we need to find the corresponding eigenvectors. For λ = 1, we have the eigenvector [1,0,1]. For λ = -2, we have the eigenvector [0,1,2].

Now, we can construct the diagonal matrix D = [[1,0,0],[0,1,-2],[0,0,-2]] and the matrix of eigenvectors P = [[1,0,0],[0,1,2],[1,0,1]].

Using the formula Y = P*D*P^-1, we can find the solution to the system Y(t) = e^(At)Y(0).

I hope this helps. Let me know if you have any further questions.
 

FAQ: How to Solve System of Equations with Matrix Y

How do I set up a system of equations with matrix Y?

To set up a system of equations with matrix Y, you first need to identify the variables and coefficients in the equations. Then, arrange the equations in a matrix form with the coefficients as the elements of the matrix and the variables as the unknowns. The last column of the matrix should contain the constants of the equations.

What is the purpose of using matrix Y to solve a system of equations?

Using matrix Y allows for a more efficient and organized way to solve a system of equations. It also allows for the use of matrix operations, such as row operations, to simplify the equations and find the solution.

How do I solve a system of equations with matrix Y?

To solve a system of equations with matrix Y, you can use the Gaussian elimination method. This involves using row operations to simplify the matrix until it is in reduced row-echelon form. The solution can then be found by back substitution.

Can I use matrix Y to solve any system of equations?

Yes, matrix Y can be used to solve any system of equations, whether it is a 2x2 system or a larger system with multiple variables. However, the method of solving may vary depending on the size and complexity of the system.

Are there any limitations to using matrix Y to solve a system of equations?

One limitation of using matrix Y to solve a system of equations is that it may not work for systems with infinitely many solutions or no solutions. In these cases, the matrix may not be able to be reduced to reduced row-echelon form, and alternative methods may need to be used.

Similar threads

Replies
1
Views
874
Replies
10
Views
1K
Replies
1
Views
1K
Replies
12
Views
2K
Replies
4
Views
1K
Replies
5
Views
1K
Replies
4
Views
2K
Replies
6
Views
3K
Back
Top