SOLUTION: Let A = [5 0.25] [35 6] Find an invertible matrix X and a diagonal matrix D such that X^−1*A*X=D.

Solve the characteristic equation by subtracting  from the main
diagonal elements:









That can actually be factored, if you notice that 2125 ends in 25,
and is divisible by 25, as 85 x 25, and it turns out that 85+25=110


 <--the eigenvalues

So the diagonal matrix D is the matrix with the eigenvalues on the main
diagonal:



To find eigenvectors, we solve



For the first eigenvalue ,





-3.5x+0.25y=0    
0.25y=3.5x
y=3.5x/0.25
y=14x

Choose x=1, and we have eigenvector 

For the second eigenvalue ,





2.5x+0.25y=0    
0.25y=-0.25x
y=-0.25x/0.25
y=-10x

Choose x=1, and we have eigenvector 

Now we combine these two eigenvectors and that gives the matrix X:



You weren't asked to find X-1. But you would need it to check the
problem, and actually to show that X is invertible. To find the inverse of X, we
would use the formula for the inverse of a 2x2 matrix:









This shows that X is invertible.  Now you have the parts you were asked for:

, 

Edwin