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Question
Find the rank of the matrix
A = `((4, 5, 2, 2),(3, 2, 1, 6),(4, 4, 8, 0))`
Solution
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
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