Find the Determinant

Problem

det(ex)

Solution

1. Factor out the common term ex from the first column to simplify the matrix.

det(A)=ex*det(1)

2. Expand along the first column using the minors of the 3×3 matrix.

det(A)=ex*(1*det(−sin(x))−1*det(cos(x))+1*det(cos(x)))

3. Calculate the 2×2 determinants for each term using the formula a*d−b*c

det(−sin(x))=(−sin(x))*(−sin(x))−(cos(x))*(−cos(x))=sin2(x)+cos2(x)=1

det(cos(x))=(cos(x))*(−sin(x))−(sin(x))*(−cos(x))=−sin(x)*cos(x)+sin(x)*cos(x)=0

det(cos(x))=(cos(x))*(cos(x))−(sin(x))*(−sin(x))=cos2(x)+sin2(x)=1

4. Substitute the values back into the expansion expression.

det(A)=ex*(1*(1)−1*(0)+1*(1))

5. Simplify the resulting expression to find the final value.

det(A)=ex*(1+1)=2*ex

Final Answer

det(ex)=2*ex

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