Exponential matrix problem and Putzer's formula

[kamenoss]

Homework Statement

e^{ \begin{bmatrix} \lambda_{1} & 0 \\ 1& \lambda_{2} \end{bmatrix}t }=\begin{bmatrix} e^{\lambda_{1}t} & 0 \\
\frac{e^{\lambda_{2}t} - e^{\lambda_{1}t}}{ \lambda_{2} - \lambda_{1}} & e^{\lambda_{2}t}
\end{bmatrix}

with \lambda_{1}\neq \lambda_{2}

Homework Equations

Could someone solve this for me?

The Attempt at a Solution

I am no good at maths... with basic knowledge of linear algebra.
It looks like \lambda_{1} , \lambda_{2} are the eigenvalues of a matrix A that solves a differential system. All indications are pointing to Putzer's formula, but everything i have tried failed. Probably i am missing something...

Thanks in advance...
p.s.Sorry for my terrible English

 

[hunt_mat]

Basically you write out the definition of the exponential:
<br /> e^{x}=1+x+\frac{x^{2}}{2!}+\cdots +\frac{x^{n}}{n!}+\cdots<br />
To compute the LHS of your equation, you will have you figure out an equation for:
<br /> \left(<br /> \begin{array}{cc}<br /> \lambda_{1} &amp; 0 \\<br /> 1 &amp; \lambda_{2}<br /> \end{array}\right)^{n}<br />
and this will give you what you require.