Question

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

`[(2,-3),(5,7)]`

Answer 1

Let A = `[(2,-3),(5,7)]`

∴ |A| = `|(2,-3),(5,7)| = 14 + 15 = 29 ne 0`

∴ A^-1 exists.

Consider AA^-1 = I

∴ `[(2,-3),(5,7)] "A"^-1 = [(1,0),(0,1)]`

By 3R₁, we get,

`[(6,-9),(5,7)] "A"^-1 = [(3,0),(0,1)]`

By R₁ - R₂, we get,

`[(1,-16),(5,7)] "A"^-1 = [(3,-1),(0,1)]`

By R₂ - 5R₁, we get,

`[(1,-16),(0,87)] "A"^-1 = [(3,-1),(-15,6)]`

By `(1/87)"R"_2,`we get

`[(1,-16),(0,1)] "A"^-1 = [(3,-1),(-5/29,2/29)]`

By R₁ + 16R₂, we get,

`[(1,0),(0,1)]"A"^-1 = [(7/29,3/29),(-5/29,2/29)]`

∴ A^-1 = `1/29 [(7,3),(-5,2)]`