Find the Inverse [[7,8],[2/3,1/3]]

Problem

[[7,8],[2/3,1/3]]

Solution

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

A=[[a,b],[c,d]]=[[7,8],[2/3,1/3]]

A(−1)=1/det(A)*[[d,−b],[−c,a]]

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

det(A)=(7)*(1/3)−(8)*(2/3)

det(A)=7/3−16/3

det(A)=−3

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

adj(A)=[[1/3,−8],[−2/3,7]]

4. Multiply the adjugate matrix by the scalar 1/det(A)

A(−1)=1/(−3)*[[1/3,−8],[−2/3,7]]

5. Simplify the resulting entries.

A(−1)=[[−1/9,8/3],[2/9,−7/3]]

Final Answer

[[7,8],[2/3,1/3]](−1)=[[−1/9,8/3],[2/9,−7/3]]

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