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Consider the matrix P = [[0, 1], [-2, -3]]. The value of e^P is

1. [[2e⁻² - 3e⁻¹, e⁻¹ - e⁻²], [2e⁻² - 2e⁻¹, 5e⁻² - e⁻¹]]
2. [[e⁻¹ + e⁻², 2e⁻² - e⁻¹], [2e⁻¹ - 4e⁻², 3e⁻¹ + 2e⁻²]]
3. [[5e⁻² - e⁻¹, 3e⁻¹ - e⁻²], [2e⁻² - 6e⁻¹, 4e⁻² + e⁻¹]]
4. [[2e⁻¹ - e⁻², e⁻¹ - e⁻²], [-2e⁻¹ + 2e⁻², -e⁻¹ + 2e⁻²]]

Correct answer: [[2e⁻¹ - e⁻², e⁻¹ - e⁻²], [-2e⁻¹ + 2e⁻², -e⁻¹ + 2e⁻²]]

Solution

Eigenvalues of P are -1 and -2. Writing e^P=c0 I+c1 P with e^-1=c0-c1 and e^-2=c0-2c1 gives c1=e^-1-e^-2, c0=2e^-1-e^-2. Then e^P=[[c0,c1],[-2c1,c0-3c1]]=[[2e^-1-e^-2, e^-1-e^-2],[-2e^-1+2e^-2, -e^-1+2e^-2]], which is option 3.

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