Find the Inverse [[46,41,18],[-14,-16,-26],[7,11,-25]]

Problem

[[46,41,18],[−14,−16,−26],[7,11,−25]]

Solution

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

|A|=46*((−16)*(−25)−(−26)*(11))−41*((−14)*(−25)−(−26)*(7))+18*((−14)*(11)−(−16)*(7))

|A|=46*(400+286)−41*(350+182)+18*(−154+112)

|A|=46*(686)−41*(532)+18*(−42)

|A|=31556−21812−756=8988

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)=686,(M_12)=532,(M_13)=−42

(M_21)=−1223,(M_22)=−1276,(M_23)=219

(M_31)=−778,(M_32)=−944,(M_33)=−162

3. Apply the cofactor signs using the pattern of (−1)(i+j) to create the cofactor matrix C

C=[[686,−532,−42],[1223,−1276,−219],[−778,944,−162]]

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

adj(A)=[[686,1223,−778],[−532,−1276,944],[−42,−219,−162]]

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

A(−1)=1/8988*[[686,1223,−778],[−532,−1276,944],[−42,−219,−162]]

Final Answer

[[46,41,18],[−14,−16,−26],[7,11,−25]](−1)=[[49/642,1223/8988,−389/4494],[−19/321,−319/2247,236/2247],[−1/214,−73/2996,−9/499]]

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