Using elementary row transformations, find the inverse of the matrix A = `[(1,2,3),(2,5,7),(-2,-4,-5)]`

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#### Solution

We know that

A = IA

i.e `[(1,2,3),(2,5,7),(-2,-4,-5)] = A[(1,0,0),(0,1,0),(0,0,1)]`

Applying R_{2}→ R_{2}−2R_{1} and R_{3}→R_{3} +2R_{1}

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

Applying R_{1}→R_{1}−2R_{2}

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

Applying R_{1}→R_{1}−R_{3}

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

Applying R_{2}→R_{2}−R_{3}

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

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

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