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

Problem

[[5*x,−4*y],[6*x,−5*y]]

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 using the given entries.

det(A)=(5*x)*(−5*y)−(−4*y)*(6*x)

det(A)=−25*x*y−(−24*x*y)

det(A)=−x*y

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

adj(A)=[[−5*y,4*y],[−6*x,5*x]]

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

A(−1)=1/(−x*y)*[[−5*y,4*y],[−6*x,5*x]]

5. Simplify the expression by distributing the scalar into the matrix.

A(−1)=[[(−5*y)/(−x*y),(4*y)/(−x*y)],[(−6*x)/(−x*y),(5*x)/(−x*y)]]

A(−1)=[[5/x,−4/x],[6/y,−5/y]]

Final Answer

[[5*x,−4*y],[6*x,−5*y]](−1)=[[5/x,−4/x],[6/y,−5/y]]

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