Find the Inverse [[-e^t,1],[e^t,e^(-t)]]

Problem

[[−et,1],[et,e(−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)=(−et)*(e(−t))−(1)*(et)

det(A)=−e(t−t)−et

det(A)=−1−et

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

adj(A)=[[e(−t),−1],[−et,−et]]

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

A(−1)=1/(−1−et)*[[e(−t),−1],[−et,−et]]

5. Simplify the expression by distributing the negative sign from the denominator into the matrix.

A(−1)=1/(1+et)*[[−e(−t),1],[et,et]]

Final Answer

[[−et,1],[et,e(−t)]](−1)=1/(1+et)*[[−e(−t),1],[et,et]]

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