Question

If A = `((2,1,3),(1,0,1),(1,1,1))`, then reduce it to I₃ by using row transformations.

Answer 1

|A| = `|(2,1,3),(1,0,1),(1,1,1)|`

= 2(0 - 1)- 1(1 - 1) + 3(1 - 0)

= - 2 - 0 + 3

= 1 ≠ 0

∴ A is a non-singular matrix.

Hence, the required transformation is possible.

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

By R₁ - R₂, we get,

A ∼ `[(1,1,2),(1,0,1),(1,1,1)]`

By R₂ - R₁ and R₃ - R₁, we get,

A ∼ `[(1,1,2),(0,-1,-1),(0,0,-1)]`

By (- 1)R₂ and (- 1)R₃, we get,

A ∼ `[(1,1,2),(0,1,1),(0,0,1)]`

By R₁ - R₂, we get,

A ∼ `[(1,0,1),(0,1,1),(0,0,1)]`

By R₁ - R₃, and R₂ - R₃, we get,

A ∼ `[(1,0,0),(0,1,0),(0,0,1)]`