Question

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

`((1,-1),(2,3))`

Answer 1

Let A = `((1,-1),(2,3))`

∴ |A| = `|(1,-1),(2,3)| = 3 + 2 = 5 ≠ 0`

∴ A^-1 exists.

Consider AA^-1 = I

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

By R₂ - 2R₁, we get,

`((1,-1),(0,5)) "A"^-1 = ((1,0),(-2,1))`

By `(1/5) "R"_2`, we get,

`((1,-1),(0,1)) "A"^-1 = ((1,0),(-2/5,1/5))`

By R₁ + R₂, we get,

`((1,0),(0,1)) "A"^-1 = ((3/5,1/5),(-2/5,1/5))`

∴ A^-1 = `1/5((3,1),(-2,1))`