How Do You Solve a Matrix Equation with Eigenvalues?

[aeronautical]

Homework Statement

Solve the matrix equation:

[10 1 9
1 4 3
2 -1 3] X = 3X

where X is a column matrix. Please show all the steps, thanks.

The Attempt at a Solution

I think the solution is X = kU, where k is any constant and U is the vector...
yet not really sure how to do this...can somebody show me all the steps?

 

[dx]

Hint: Let A be the matrix. Then you want to solve Ax = 3x. This is equivalent to (A-3I)x = 0, where I is the identity matrix. Write it in this form and it will be easier to see a solution.