Find the Inverse [[7,4,5],[2,1,3],[7,8,9]]

Problem

[[7,4,5],[2,1,3],[7,8,9]](−1)

Solution

1. Calculate the determinant of the matrix A using the first row.

det(A)=7*(1⋅9−3⋅8)−4*(2⋅9−3⋅7)+5*(2⋅8−1⋅7)

det(A)=7*(9−24)−4*(18−21)+5*(16−7)

det(A)=7*(−15)−4*(−3)+5*(9)

det(A)=−105+12+45=−48

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_11)=9−24=−15,(M_12)=18−21=−3,(M_13)=16−7=9

(M_21)=36−40=−4,(M_22)=63−35=28,(M_23)=56−28=28

(M_31)=12−5=7,(M_32)=21−10=11,(M_33)=7−8=−1

3. Apply the cofactor signs using the pattern +−+ on the matrix of minors.

C=[[−15,3,9],[4,28,−28],[7,−11,−11]]

4. Transpose the cofactor matrix to find the adjugate matrix adj(A)

adj(A)=[[−15,4,7],[3,28,−11],[9,−28,−1]]

5. Multiply by the reciprocal of the determinant to find the inverse matrix A(−1)=1/det(A)*adj(A)

A(−1)=1/(−48)*[[−15,4,7],[3,28,−11],[9,−28,−1]]

Final Answer

[[7,4,5],[2,1,3],[7,8,9]](−1)=[[5/16,−1/12,−7/48],[−1/16,−7/12,11/48],[−3/16,7/12,1/48]]

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