Find the Inverse [[1/2,1],[1,1]]

Problem

[[1/2,1],[1,1]]

Solution

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

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

A(−1)=1/det(A)*[[d,−b],[−c,a]]

2. Calculate the determinant of the matrix.

det(A)=(1/2)*(1)−(1)*(1)

det(A)=1/2−1

det(A)=−1/2

3. Apply the formula by swapping the diagonal elements and changing the signs of the off-diagonal elements.

A(−1)=1/(−1/2)*[[1,−1],[−1,1/2]]

4. Simplify the expression by multiplying the matrix by the scalar −2

A(−1)=−2*[[1,−1],[−1,1/2]]

A(−1)=[[−2,2],[2,−1]]

Final Answer

[[1/2,1],[1,1]](−1)=[[−2,2],[2,−1]]

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