Calculating the Exponential of a 2x2 Matrix with a Variable

[clayy]

¡¡get matrix exponential Please!

Homework Statement

I have a exam and i don't know how get matrix exponential:

| 2*t t|
| 3*t -t|
it is a 2x2 matrix.
where 't' is not a constant ,it is a variable
somehere could help me,please.

Homework Equations

The Attempt at a Solution

Homework Statement

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

If A is a diagonal matrix with only 0 off the diagonal, a_ii on the diagonal, then e^A is a diagonal matrix with e^aii on the diagonal. If A is a "diagonalizable" matrix, that is, if there exist a matrix P such that PAP^-1= D, a diagonal matrix, then $e^A= P^{-1}e^DP$.

If A is not diagonalizable, then you can still put it in "Jordan Normal Form", which is "near diagonal", but the exponential is more complicated.

Fortunately for you, this particular matrix has two distinct eigenvalues for every t (except 0 in which case the matrix is identically 0 and so its exponential is the identity matrix) and so is diagonalizable. Find the eigenvalues and corresponding eigenvectors. The exponential will be P^-1DP where P is the matrix having the eigenvectors as columns and D is the diagonal matrix with $e^{\lambda}$ on the diagonal with $\lambda$ being the eigenvalues.

 

[clayy]

Thank you very much.