Find the Inverse [[19,11,21],[9,0,10],[10,12,11]]

Problem

[[19,11,21],[9,0,10],[10,12,11]](−1)

Solution

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

det(A)=−9*(11⋅11−21⋅12)+0−10*(19⋅12−11⋅10)

det(A)=−9*(121−252)−10*(228−110)

det(A)=−9*(−131)−10*(118)

det(A)=1179−1180=−1

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)=(0)*(11)−(10)*(12)=−120

(M_12)=(9)*(11)−(10)*(10)=−1

(M_13)=(9)*(12)−(0)*(10)=108

(M_21)=(11)*(11)−(21)*(12)=−131

(M_22)=(19)*(11)−(21)*(10)=−1

(M_23)=(19)*(12)−(11)*(10)=118

(M_31)=(11)*(10)−(21)*(0)=110

(M_32)=(19)*(10)−(21)*(9)=1

(M_33)=(19)*(0)−(11)*(9)=−99

3. Apply the cofactor signs using the pattern +−+ −+− +−+ to create the cofactor matrix C

C=[[−120,1,108],[131,−1,−118],[110,−1,−99]]

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

adj(A)=[[−120,131,110],[1,−1,−1],[108,−118,−99]]

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

A(−1)=1/(−1)*[[−120,131,110],[1,−1,−1],[108,−118,−99]]

A(−1)=[[120,−131,−110],[−1,1,1],[−108,118,99]]

Final Answer

[[19,11,21],[9,0,10],[10,12,11]](−1)=[[120,−131,−110],[−1,1,1],[−108,118,99]]

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