Calculator Form
Choose a square matrix size, enter values, and calculate invertibility instantly.
Formula Used
Invertibility rule: A square matrix A is invertible if and only if det(A) ≠ 0.
Rank rule: A square matrix is invertible when rank(A) = n.
2 × 2 shortcut: For A = [[a, b], [c, d]], det(A) = ad − bc.
Elimination rule: After row reduction, det(A) = (−1)s × product of pivots, where s is the number of row swaps.
The chart uses the scaling identity det(kA) = kn det(A), where n is the matrix size.
How to Use This Calculator
- Select the matrix size from 2 × 2 through 5 × 5.
- Enter every matrix entry in the grid.
- Click Determine Invertibility to run the calculation.
- Read the determinant, rank, and inverse status.
- Review the determinant and inverse-check steps.
- Use the CSV or PDF buttons to save results.
- Study the graph to see determinant scaling behavior.
Example Data Table
| Example | Matrix | Determinant | Rank | Invertible |
|---|---|---|---|---|
| Identity 2 × 2 | [1 0; 0 1] | 1 | 2 | Yes |
| Repeated Rows | [1 2; 1 2] | 0 | 1 | No |
| Classic 3 × 3 | [1 2 3; 0 1 4; 5 6 0] | 1 | 3 | Yes |
| Dependent 3 × 3 | [2 4 6; 1 2 3; 3 6 9] | 0 | 1 | No |
FAQs
1. What makes a matrix invertible?
A square matrix is invertible when its determinant is not zero. Equivalent tests also work, such as full rank and independent rows or columns.
2. Why must the matrix be square?
Only square matrices can have true two-sided inverses. Non-square matrices may have related concepts, but not a standard inverse matrix.
3. Is determinant alone enough?
Yes, for square matrices. If det(A) ≠ 0, the matrix is invertible. If det(A) = 0, the matrix is singular and not invertible.
4. Why does rank matter here?
Rank confirms how many independent rows or columns exist. For an n × n matrix, rank must equal n for invertibility.
5. What does a zero determinant mean?
A zero determinant means the matrix compresses space into a lower dimension. That causes information loss, so no inverse can recover original vectors.
6. Does row swapping change invertibility?
No. Row swapping changes the determinant sign, not whether the determinant is zero. Invertibility stays the same after a swap.
7. Why show the inverse matrix too?
Showing the inverse helps verify the result and supports follow-up work. You can directly use A⁻¹ for solving systems or checking products.
8. What size matrices does this page support?
This version supports square matrices from 2 × 2 through 5 × 5. That range keeps the page quick, readable, and practical.