Matrix Exponentials

Practice Questions

Differential Equations › Matrix Exponentials

Questions

6

1

Find the general solution to

$y^{'''} + y^{''} + 3y' - 5y = 0$

2

Use the definition of matrix exponential,

$e^{At}=I+At+A^2\frac{t^2}{2!}+...+A^k\frac{t^k}{k!}+...=\sum_{k=0}^\infty A^k\frac{t^k}{k!}$

to compute $e^{At}$ of the following matrix.

$A=\begin{pmatrix} 0& 1\ 2&1 \end{pmatrix}$

3

Use the definition of matrix exponential,

$e^{At}=I+At+A^2\frac{t^2}{2!}+...+A^k\frac{t^k}{k!}+...=\sum_{k=0}^\infty A^k\frac{t^k}{k!}$

to compute $e^{At}$ of the following matrix.

$A=\begin{pmatrix} 0& 1\ 2&1 \end{pmatrix}$

4

Given the matrix $A =\begin{bmatrix} -1 & -1& -3 \ -6& 1& 0 \ 0& 1 &2 \end{bmatrix}$ , calculate the matrix exponential, $e^{tA}$ . You may leave your answer diagonalized: i.e. it may contain matrices multiplied together and inverted.

5

Given the matrix $A =\begin{bmatrix} 3 & 1 \ 1& 3 \end{bmatrix}$ , calculate the matrix exponential, $e^{tA}$ .

6

Calculate the matrix exponential, $e^{tA}$ , for the following matrix: $A = \begin{bmatrix} -7& 3\-1 &-3 \end{bmatrix}$ .

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