Find the Inverse [[3,-2,3],[5,0,4],[2,7,0]]

Problem

[[3,−2,3],[5,0,4],[2,7,0]]

Solution

1. Calculate the determinant of the matrix using the cofactor expansion along the first row.

det(A)=3*(0⋅0−4⋅7)−(−2)*(5⋅0−4⋅2)+3*(5⋅7−0⋅2)

det(A)=3*(−28)+2*(−8)+3*(35)

det(A)=−84−16+105=5

2. Find the matrix of minors by calculating the determinant of the 2×2 matrix remaining after removing the row and column of each element.

M=[[−28,−8,35],[−21,−6,25],[−8,−3,10]]

3. Apply the cofactor signs using the pattern of alternating signs (+−+)

C=[[−28,8,35],[21,−6,−25],[−8,3,10]]

4. Transpose the cofactor matrix to find the adjugate matrix.

adj(A)=CT=[[−28,21,−8],[8,−6,3],[35,−25,10]]

5. Multiply by the reciprocal of the determinant to find the inverse matrix.

A(−1)=1/5*[[−28,21,−8],[8,−6,3],[35,−25,10]]

Final Answer

[[3,−2,3],[5,0,4],[2,7,0]](−1)=[[−5.6,4.2,−1.6],[1.6,−1.2,0.6],[7,−5,2]]

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