Find the Determinant

Problem

det(−3*e^(2*t))

Solution

1. Identify the formula for the determinant of a 2×2 matrix [[a,b],[c,d]] which is a*d−b*c

2. Substitute the elements of the matrix into the formula: a=−3*e^(2*t) b=−4*e^(3*t) c=−6*e^(2*t) and d=−3*e^(3*t)

det(−3*e^(2*t))=(−3*e^(2*t))*(−3*e^(3*t))−(−4*e^(3*t))*(−6*e^(2*t))

3. Multiply the terms using the property of exponents e^u⋅e^v=e^(u+v)

(−3*e^(2*t))*(−3*e^(3*t))=9*e^(5*t)

(−4*e^(3*t))*(−6*e^(2*t))=24*e^(5*t)

4. Subtract the resulting terms to find the final value.

9*e^(5*t)−24*e^(5*t)=−15*e^(5*t)

Final Answer

det(−3*e^(2*t))=−15*e^(5*t)

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