Find the Inverse [[7,3],[-6,-3]]

Problem

[[7,3],[−6,−3]]

Solution

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

A=[[a,b],[c,d]]=[[7,3],[−6,−3]]

A(−1)=1/(a*d−b*c)*[[d,−b],[−c,a]]

2. Calculate the determinant of the matrix, which is a*d−b*c

det(A)=(7)*(−3)−(3)*(−6)

det(A)=−21−(−18)

det(A)=−3

3. Swap the elements on the main diagonal and change the signs of the elements on the off-diagonal.

[[d,−b],[−c,a]]=[[−3,−3],[6,7]]

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

A(−1)=1/(−3)*[[−3,−3],[6,7]]

5. Simplify the matrix by distributing the scalar multiplication.

A(−1)=[[(−3)/(−3),(−3)/(−3)],[6/(−3),7/(−3)]]

A(−1)=[[1,1],[−2,−7/3]]

Final Answer

[[7,3],[−6,−3]](−1)=[[1,1],[−2,−7/3]]

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