Question

exponential of a matrix.

i'm trying to compute \(e^{tA}\)
\[A=\left[\begin{matrix} \lambda & 0 & 0 \\ 1 & \lambda & 0 \\ 0 & 1 & \lambda \end{matrix}\right]\]

Answer 1

so I know that \[(ta)^k=\left[\begin{matrix} (t\lambda)^k & 0 & 0 \\ t^k & (t\lambda)^k & 0 \\ 0 & t^k & (t\lambda)^k \end{matrix}\right]\]

Answer 2

and can figure out all the non-zero parts. why do the zeroes end up being one though?

Answer 3

Diagonalize the matrix