Find the Inverse [[-6x+7y,-25],[-4x-5y,-7]]

Problem

[[−6*x+7*y,−25],[−4*x−5*y,−7]](−1)

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)=(−6*x+7*y)*(−7)−(−25)*(−4*x−5*y)

det(A)=42*x−49*y−(100*x+125*y)

det(A)=42*x−49*y−100*x−125*y

det(A)=−58*x−174*y

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

adj(A)=[[−7,25],[4*x+5*y,−6*x+7*y]]

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

A(−1)=1/(−58*x−174*y)*[[−7,25],[4*x+5*y,−6*x+7*y]]

Final Answer

[[−6*x+7*y,−25],[−4*x−5*y,−7]](−1)=[[7/(58*x+174*y),(−25)/(58*x+174*y)],[(−4*x−5*y)/(58*x+174*y),(6*x−7*y)/(58*x+174*y)]]

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