Question

Find the inverse of the following matrix (if they exist):

`((2,1),(1,-1))`

Answer 1

Let A = `((2,1),(1,-1))`

∴ |A| = `|(2,1),(1,-1)| = - 2 - 1 = - 3 ≠ 0`

∴ A^-1 exists.

Consider AA^-1 = I

∴ `((2,1),(1,-1)) "A"^-1 = ((1,0),(0,1))`

By R₁ ↔ R₂, we get,

`((1,-1),(2,1)) "A"^-1 = ((0,1),(1,0))`

By R₂ - 2R₁, we get,

`((1,-1),(0,3)) "A"^-1 = ((0,1),(1,-2))`

By `(1/3) "R"_2`, we get,

`((1,-1),(0,1)) "A"^-1 = ((0,1),(1/3,-2/3))`

By R₁ + R₂, we get,

`((1,0),(0,1)) "A"^-1 = ((1/3,1/3),(1/3,-2/3))`

∴ A^-1 = `1/3((1,1),(1,-2))`