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?

Answer 1

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.