Find the Determinant

Problem

det(e(−3*x)*cos(2*x))

Solution

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

2. Set up the product a*d using the given entries.

a*d=(e(−3*x)*cos(2*x))*(−3*e(−3*x)*sin(2*x)+2*e(−3*x)*cos(2*x))

3. Distribute the terms in the product a*d

a*d=−3*e(−6*x)*cos(2*x)*sin(2*x)+2*e(−6*x)*cos2(2*x)

4. Set up the product b*c using the given entries.

b*c=(e(−3*x)*sin(2*x))*(−3*e(−3*x)*cos(2*x)−2*e(−3*x)*sin(2*x))

5. Distribute the terms in the product b*c

b*c=−3*e(−6*x)*sin(2*x)*cos(2*x)−2*e(−6*x)*sin2(2*x)

6. Subtract b*c from a*d to find the determinant.

det(=)*(−3*e(−6*x)*cos(2*x)*sin(2*x)+2*e(−6*x)*cos2(2*x))−(−3*e(−6*x)*sin(2*x)*cos(2*x)−2*e(−6*x)*sin2(2*x))

7. Simplify the expression by canceling the identical terms −3*e(−6*x)*cos(2*x)*sin(2*x) and +3*e(−6*x)*sin(2*x)*cos(2*x)

det(=)*2*e(−6*x)*cos2(2*x)+2*e(−6*x)*sin2(2*x)

8. Factor out the common term 2*e(−6*x)

det(=)*2*e(−6*x)*(cos2(2*x)+sin2(2*x))

9. Apply the Pythagorean identity cos2(θ)+sin2(θ)=1

det(=)*2*e(1)(−6*x)

Final Answer

det(e(−3*x)*cos(2*x))=2*e(−6*x)

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