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Question
Apply the given elementary transformation of the following matrix.
A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R3 and then C3 + 2C2
and A = `[(1,-1,3),(2,1,0),(3,3,1)]`, C3 + 2C2 and then 3R3
What do you conclude?
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Solution
A = `[(1,-1,3),(2,1,0),(3,3,1)]`
By 3R3, we get,
A ∼ `[(1,-1,3),(2,1,0),(9,9,3)]`
By C3 + 2C2, we get,
A ∼ `[(1,-1,3+2(-1)),(2,1,+2(1)),(9,9,+2(9))]`
∴ A ∼ `[(1,-1,1),(2,1,2),(9,9,21)]` ..............(i)
And
A = `[(1,-1,3),(2,1,0),(3,3,1)]`
By C3 + 2C2, we get,
A ∼ `[(1,-1,3+2(-1)),(2,1,0+2(1)),(3,3,+1+2(3))]`
∴ A ∼ `[(1,-1,1),(2,1,2),(3,3,7)]`
∴ A ∼ `[(1,-1,1),(2,1,2),(3,3,7)]`
By 3R3, we get
A ∼ `[(1,-1,1),(2,1,2),(9,9,21)]` ......(ii)
We conclude from (i) and (ii) the matrix remains the same by interchanging the order of the elementary transformations. Hence, the transformations are commutative.
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