Find the Inverse [[-1/4,-1/6],[1/3,4/3]]

Problem

[[−1/4,−1/6],[1/3,4/3]]

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)=(−1/4)*(4/3)−(−1/6)*(1/3)

det(A)=−1/3+1/18

det(A)=−6/18+1/18=−5/18

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

adj(A)=[[4/3,1/6],[−1/3,−1/4]]

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

A(−1)=−18/5*[[4/3,1/6],[−1/3,−1/4]]

5. Distribute the scalar across the matrix elements.

A(−1)=[[(−18/5)*(4/3),(−18/5)*(1/6)],[(−18/5)*(−1/3),(−18/5)*(−1/4)]]

A(−1)=[[−24/5,−3/5],[6/5,9/10]]

Final Answer

[[−1/4,−1/6],[1/3,4/3]](−1)=[[−24/5,−3/5],[6/5,9/10]]

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