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The state transition matrix for the system [ẋ1; ẋ2] = [[1,0],[1,1]] [x1; x2] + [1;1] u is

1. [e^t 0; e^t e^t]
2. [e^t 0; t² e^t e^t]
3. [e^t 0; t e^t e^t]
4. [e^t t e^t; 0 e^t]

Correct answer: [e^t 0; t e^t e^t]

Solution

The correct option represents the state transition matrix derived from the system's dynamics, where the first row reflects the exponential growth of the first state and the second row captures the interaction between the states over time, specifically incorporating the effect of the input through the term 't e^t'.

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