$$\left[ {\matrix{ 0 & {{e^{ - t}}} \cr {{0^{ - t}}} & 0 \cr } } \right]$$

$$\left[ {\matrix{ {{e^t}} & 0 \cr 0 & {{e^t}} \cr } } \right]$$

$$\left[ {\matrix{ {{e^{ - t}}} & 0 \cr 0 & {{e^{ - t}}} \cr } } \right]$$

$$\left[ {\matrix{ 0 & {{e^t}} \cr {{e^t}} & 0 \cr } } \right]$$

$${1 \over 3}\left[ {\matrix{ {\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} & { - 2\sin \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( { - t} \right)} & {2\sin \left( { - 4t} \right) + \sin \left( { - t} \right)} \cr } } \right]$$

$$\left[ {\matrix{ {\sin \left( { - 2t} \right)} & {\sin \left( {2t} \right)} \cr {\sin \left( t \right)} & {\sin \left( { - 3t} \right)} \cr } } \right]$$

$${1 \over 3}\left[ {\matrix{ {\sin \left( {4t} \right) + 2\sin \left( t \right)} & {2\sin \left( { - 4t} \right) - 2\sin \left( { - t} \right)} \cr { - \sin \left( { - 4t} \right) + \sin \left( t \right)} & {2\sin \left( {4t} \right) + \sin \left( t \right)} \cr } } \right]$$

$${1 \over 3}\left[ {\matrix{ {\cos \left( { - t} \right) + 2\cos \left( t \right)} & {2\cos \left( { - 4t} \right) + 2\sin \left( { - t} \right)} \cr { - \cos \left( { - 4t} \right) + \sin \left( { - t} \right)} & { - 2\cos \left( { - 4t} \right) + \cos \left( { - t} \right)} \cr } } \right]$$

controllable but not observable.

observable but not controllable.

neither controllable nor observable.

controllable and observable.

$${{\left( {s + 10} \right)} \over {\left( {s + 1} \right)\left( {s + 100} \right)}}$$

$${{10\left( {s + 1} \right)} \over {\left( {s + 10} \right)\left( {s + 100} \right)}}$$

$${{{{10}^2}\left( {s + 1} \right)} \over {\left( {s + 10} \right)\left( {s + 100} \right)}}$$

$${{{{10}^3}\left( {s + 100} \right)} \over {\left( {s + 1} \right)\left( {s + 10} \right)}}$$