Question

Find the rank of the matrix

A = `((4, 5, 2, 2),(3, 2, 1, 6),(4, 4, 8, 0))`

Answer 1

A = `[(4, 5, 2, 2),(3, 2, 1, 6),(4, 4, 8, 0)]`

The order of A is 3 × 4

∴ p(A) < 3

Let us transform the matrix A to an echelon form

Martix	Elementary Transformation
A = `[(4, 5, 2, 2),(3, 2, 1, 6),(4, 4, 8, 0)]`
`∼ [(3, 2, 1, 6),(4, 5, 2, 2),(4, 4, 8, 0)]`	`{:"R"_1 ↔ "R"_2:}`
`∼ [(3, 2, 1, 6),(4, 5, 2, 2),(0, -1, 6, -2)]`	`{:"R"_3 ↔ "R"_2:}`
`∼ [(3, 2, 1, 6),(1, 3, 1, -4),(0, -1, 6, -2)]`	`{:"R"_2 -> "R"_2 - "R"_1:}`
`∼ [(1, 3, 1, -4),(3, 2, 1, 6),(0, -1, 6, -2)]`	`{:"R"_1 ↔ "R"_2:}`
`∼ [(1, 3, 1, -4),(0, -7, -2, 18),(0, -1, 6, -2)]`	`{:"R"_2 -> "R"_3 - 3"R"_1:}`
`∼ [(1, 3, 1, -4),(0, -1, 6, -2),(0, -7, -2, 18)]`	`{:"R"_2 ↔ "R"_3:}`
`∼ [(1, 3, 1, -4),(0, -1, 6, -2),(0, 0, -44, 32)]`	`{:"R"_3 -> "R"_3 - 3"R"_1:}`

The last equivalent matrix is in the echelon form.

Number of non-zero rows = 3

∴ p(A) = 3