Question

Find the rank of the following matrices by row reduction method:

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

Answer 1

A = `[(1, 2, -1),(3, -1, 2),(1, -2, 3),(1, -1, 1)]`

`{:("R"_2 -> "R"_2 - 3"R"_1),("R"_3 -> "R"_3 - "R"_1),("R"_4 -> "R"_4 - "R"_1),(->):} [(1, 2, -1),(0, -7, 5),(0, -4, 4),(0, -3, 2)]`

`{:("R"_2 -> (- 1)"R"_2),("R"_3 -> (- 1)"R"_3),("R"_4 -> (-1)"R"_4),(->):} [(1, 2, -1),(0, 7, -5),(0, -4, 4),(0, -3, 2)]`

`{:("R"_3 -> 7"R"_3 - 4"R"_2),("R"_4 -> 7"R"_4 - 3"R"_2),(->):} [(1, 2, -1),(0, 7, -5),(0, 0, -8),(0, 0, 1)]`

`{:("R"_3 -> ("R"_3)/(-8)),(->):} [(1, 2, -1),(0, 7, -5),(0, 0, 1),(0, 0, 1)]`

`{:("R"_4 -> "R"_4 - "R"_3),(->):} [(1, 2, -1),(0, 7, -5),(0, 0, 1),(0, 0, 0)]`

The last equivalent matrix is in row echelon form.

It has three non-zero rows.

∴ P(A) = 3