Find the Inverse [[21,25],[25,35]]

Problem

[[21,25],[25,35]]

Solution

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

A=[[a,b],[c,d]]=[[21,25],[25,35]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

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

det(A)=(21)*(35)−(25)*(25)

det(A)=735−625

det(A)=110

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

adj(A)=[[35,−25],[−25,21]]

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

A(−1)=1/110*[[35,−25],[−25,21]]

5. Simplify the resulting fractions by dividing each term by the common factor where possible.

A(−1)=[[35/110,(−25)/110],[(−25)/110,21/110]]

A(−1)=[[7/22,−5/22],[−5/22,21/110]]

Final Answer

[[21,25],[25,35]](−1)=[[7/22,−5/22],[−5/22,21/110]]

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