Find the Inverse [[3e^t,e^(2t)],[2e^t,2e^(2t)]]

Problem

[[3*e^t,e^(2*t)],[2*e^t,2*e^(2*t)]]

Solution

1. Identify the matrix A and the formula for the inverse of a 2×2 matrix.

A=[[a,b],[c,d]]

A^(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

2. Calculate the determinant det(A)=a*d−b*c

det(A)=(3*e^t)*(2*e^(2*t))−(e^(2*t))*(2*e^t)

det(A)=6*e^(3*t)−2*e^(3*t)

det(A)=4*e^(3*t)

3. Construct the adjugate matrix by swapping the diagonal elements and changing the signs of the off-diagonal elements.

adj(A)=[[2*e^(2*t),−e^(2*t)],[−2*e^t,3*e^t]]

4. Multiply the adjugate matrix by the reciprocal of the determinant.

A^(−1)=1/(4*e^(3*t))*[[2*e^(2*t),−e^(2*t)],[−2*e^t,3*e^t]]

5. Simplify each term in the matrix by dividing by 4*e^(3*t) using the laws of exponents.

(2*e^(2*t))/(4*e^(3*t))=1/(2*e^t)=1/2*e^(−t)

(−e^(2*t))/(4*e^(3*t))=−1/(4*e^t)=−1/4*e^(−t)

(−2*e^t)/(4*e^(3*t))=−1/(2*e^(2*t))=−1/2*e^(−2*t)

(3*e^t)/(4*e^(3*t))=3/(4*e^(2*t))=3/4*e^(−2*t)

Final Answer

[[3*e^t,e^(2*t)],[2*e^t,2*e^(2*t)]]^(−1)=[[1/2*e^(−t),−1/4*e^(−t)],[−1/2*e^(−2*t),3/4*e^(−2*t)]]

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