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Find the Inverse of the Following Matrix, Using Elementary Transformations: a = ⎡ ⎢ ⎣ 2 3 1 2 4 1 3 7 2 ⎤ ⎥ ⎦ - Mathematics

Sum

Find the inverse of the following matrix, using elementary transformations: 

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

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Solution

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]` 

AA-1 =I 

`[[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]] A^(-1) = [[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,1]]` 

R2 → R2 - R1

R3 → R3 - R1

`[[2 , 3 , 1 ],[0 , 1 , 0],[1, 4 ,1]] A^(-1) = [[1 , 0 , 0 ],[-1 , 1 , 0],[-1 , 0 ,1]]` 

R1 ↔ R

`[[1 , 4 , 1 ],[0 , 1 , 0],[2 , 3 ,1]] A^(-1) = [[-1 , 0 , 1 ],[-1 , 1 , 0],[1 , 0 ,0]]` 

R3 → R3 - 2R1

`[[1 , 4 , 1 ],[0 , 1 , 0],[0 , -5 ,-1]] A^(-1) = [[-1 , 0 , 1 ],[-1 , 1 , 0],[3 , 0 ,-2]]` 

R1 → R1 - 4R2

R3 → R3 - 5R2

`[[1 , 0 , 1 ],[0 , 1 , 0],[0 , 0 ,-1]] A^(-1) = [[3 , -4 , 1 ],[-1 , 1 , 0],[-2 , 5 ,-2]]` 

`R_1 ->R_1 +R_3`

`[[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,-1]] A^(-1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[-2 , 5 ,-2]]` 

`R_3 -> -R_3`

`[[1 , 0 , 0 ],[0 , 1 , 0],[0 , 0 ,1]] A^(-1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]` 

`I .A^(1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]` 

` ⇒A^(1) = [[1 , 1 , -1 ],[-1 , 1 , 0],[2 , -5 ,2]]` 

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