Question

8. Combine Exercises 4 and 7 and the definition (4.3) to find $e^A$ where A = $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (a) $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ (b) $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$ (c) 4. Show that if $A = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{bmatrix}$, $e^A = \begin{bmatrix} e^{\lambda_1} & 0 & 0 \\ 0 & e^{\lambda_2} & 0 \\ 0 & 0 & e^{\lambda_3} \end{bmatrix}$. 7. If A and B are n x n matrices such that AB = BA, show that $e^Ae^B = e^{A+B}$. The sum of this series is denoted $e^A$, that is, $e^A = \sum_{k=0}^\infty \frac{A^k}{k!}$. (4.3)

          8. Combine Exercises 4 and 7 and the definition (4.3) to find $e^A$ where A = 
$\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (a) $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$ (b) $\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{bmatrix}$ (c)
4. Show that if $A = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{bmatrix}$, $e^A = \begin{bmatrix} e^{\lambda_1} & 0 & 0 \\ 0 & e^{\lambda_2} & 0 \\ 0 & 0 & e^{\lambda_3} \end{bmatrix}$.
7. If A and B are n x n matrices such that AB = BA, show that $e^Ae^B = e^{A+B}$.
The sum of this series is denoted $e^A$, that is,
$e^A = \sum_{k=0}^\infty \frac{A^k}{k!}$.
(4.3)

8. Combine Exercises 4 and 7 and the definition (4.3) to find e^A where A =
< b m a t r i x > (a) < b m a t r i x > (b) < b m a t r i x > (c)
4. Show that if A =
< b m a t r i x >, e^A =
< b m a t r i x >.
7. If A and B are n x n matrices such that AB = BA, show that e^Ae^B = e^A+B.
The sum of this series is denoted e^A, that is,
e^A = ∑k=0^∞(A^k)/(k!).
(4.3)

Added by Andr-S D.

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