Question

Find the rank of the matrix

A = `((-2, 1, 3, 4),(0, 1, 1, 2),(1, 3, 4, 7))`

Answer 1

A = `[(-2, 1, 3, 4),(0, 1, 1, 2),(1, 3, 4, 7)]`

The order of A is 3 × 4

∴ P(A) < 3

Let us transform the matrix A to an echelon form

Matrix	Elementary Transformation
A = `[(-2, 1, 3, 4),(0, 1, 1, 2),(1, 3, 4, 7)]`
`∼ [(-2, 1, 3, 4),(1, 3, 4, 7),(0, 1, 1, 2)]`	`{:"R"_2 ↔ "R"_3:}`
`∼ [(1, 3, 4, 7),(-2, 1, 3, 4),(0, 1, 1, 2)]`	`{:"R"_2 ↔ "R"_1:}`
`∼  [(1, 3, 4, 7),(0, 7, 11, 18),(0, 1, 1, 2)]`	`{:"R"_2 -> "R"_2 + 2"R"_1:}`
`∼ [(1, 3, 4, 7),(0, 1, 1, 2),(0, 7, 11, 18)]`	`{:"R"_3 ↔ "R"_2:}`
`∼ [(1, 3, 4, 7),(0, 1, 1, 2),(0, 0, 4, 4)]`	`{:"R"_3 -> "R"_3 + 7"R"_2:}`

The number of non-zero rows = 3

∴ p(A) = 3