Calculate the rank of a 2×2 matrix with complex numbers online for free using our accurate Rank of Matrix Calculator.
Matrix (3×3)
A1
A2
A3
B1
B2
B3
C1
C2
C3
Matrix Rank
3
| Step | Description |
|---|---|
| 1 | Input matrix is [[1+i, 2, 3], [4, 5-i, 6], [7, 8, 9-i]]. |
| 2 | Apply Gaussian elimination to reduce to row echelon form. |
| 3 | Matrix: [[1+i, 2, 3], [4, 5-i, 6], [7, 8, 9-i]]. First row pivot: 1+i. |
| 4 | Eliminate below pivot: Row 2 = Row 2 – (4/(1+i)) * Row 1. |
| 5 | Compute multiplier: 4/(1+i) = (4(1-i))/(1+i)(1-i) = (4-4i)/2 = 2-2i. |
| 6 | Row 2 = [4, 5-i, 6] – (2-2i) * [1+i, 2, 3] = [0, 1+i, 0]. |
| 7 | Eliminate below pivot: Row 3 = Row 3 – (7/(1+i)) * Row 1. |
| 8 | Compute multiplier: 7/(1+i) = (7(1-i))/(1+i)(1-i) = (7-7i)/2 = 3.5-3.5i. |
| 9 | Row 3 = [7, 8, 9-i] – (3.5-3.5i) * [1+i, 2, 3] = [0, -3+4i, 0.5-0.5i]. |
| 10 | Matrix: [[1+i, 2, 3], [0, 1+i, 0], [0, -3+4i, 0.5-0.5i]]. Second row pivot: 1+i. |
| 11 | Eliminate below pivot: Row 3 = Row 3 – ((-3+4i)/(1+i)) * Row 2. |
| 12 | Compute multiplier: (-3+4i)/(1+i) = ((-3+4i)(1-i))/(1+i)(1-i) = (1+7i)/2 = 0.5+3.5i. |
| 13 | Row 3 = [0, -3+4i, 0.5-0.5i] – (0.5+3.5i) * [0, 1+i, 0] = [0, 0, 0.5-0.5i]. |
| 14 | Row echelon form: [[1+i, 2, 3], [0, 1+i, 0], [0, 0, 0.5-0.5i]]. |
| 15 | Count non-zero rows: 3 rows are non-zero. |
| Final | Rank of the matrix is 3. |
Please enter valid complex numbers (e.g., “3+2i”, “-1-i”, “5”, “i”, “-i”) in all fields.
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