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Question
Find the inverse of the following by Gauss-Jordan method:
`[(1, 2, 3),(2, 5, 3),(1, 0, 8)]`
Solution
A = `[(1, 2, 3),(2, 5, 3),(1, 0, 8)]`
`[A|I3] = `[(1, 2, 3, |, 1, 0, 0),(2, 5, 3, |, 0, 1, 0),(1, 0, 8, |, 0, 0, 1)]`
`{:("R"_2 -> "R"_ - 2"R"_1),("R"_3 -> "R"_3 - "R"_1),(->):} [(1, 2, 3, |, 1, 0, 0),(0, 1, -3, |, -2, 1, 0),(0, -2, -5, |, -1, 0, 1)]`
`{:("R"_3 -> "R"_3 + 2"R"_2),(->):} [(1, 2, 3, |, 1, 0, 0),(0, 1, -3, |, -2, 1, 0),(0, 0, -1, |, -5, 2, 1)]`
`{:("R"_2 -> (-) "R"_3),(->):} [(1, 2, 3, |, 1, 0, 0),(0, 1, -3, |, -2, 1, 0),(0, 0, 1, |, 5, -2, -1)]`
`{:("R"_2 -> "R"_2 + 3"R"_3),(->):} [(1, 2, 3, |, 1, 0, 0),(0, 1, -3, |, -2, 1, 0),(0, 0, 1, |, 5, -2, -1)]`
`{:("R"_2 -> "R"_2 + "R"_1),(->):} [(1, 2, 3, |, 1, 0, 0),(0, 1, 0, |, 13, -5, -3),(0, 0, 1, |, 5, -2, -1)]`
`{:("R"_1 -> "R"_1 - 2"R"_2),(->):} [(1, 0, 3, |, -25, 10, 6),(0, 1, 0, |, 13, -5, -3),(0, 0, 1, |, 5, -2, -1)]`
`{:("R"_1 -> "R"_1 - 3"R"_3),(->):} [(1, 0, 0, |, -40, 16, 9),(0, 1, 0, |, 13, -5, -3),(0, 0, 1, |, 5, -2, -1)]`
∴ `"A"^-1 = [(-40, 16, 9),(13, -5, -3),(5, -2, -1)]`
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