Find the Inverse [[0.2,0.3],[0.8,0.7]]

Problem

[[0.2,0.3],[0.8,0.7]]

Solution

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

A=[[a,b],[c,d]]=[[0.2,0.3],[0.8,0.7]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(0.2)*(0.7)−(0.3)*(0.8)

det(A)=0.14−0.24

det(A)=−0.1

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

adj(A)=[[0.7,−0.3],[−0.8,0.2]]

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

A(−1)=1/(−0.1)*[[0.7,−0.3],[−0.8,0.2]]

A(−1)=−10*[[0.7,−0.3],[−0.8,0.2]]

5. Simplify the resulting matrix by performing scalar multiplication.

A(−1)=[[−7,3],[8,−2]]

Final Answer

[[0.2,0.3],[0.8,0.7]](−1)=[[−7,3],[8,−2]]

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