Build an Identity Matrix
Choose the order, diagonal value, label, and display precision. Press submit to generate the matrix and its core properties.
Example Data Table
| Example | Order | Diagonal Value | Trace | Determinant | Rank |
|---|---|---|---|---|---|
| Standard identity | 3 | 1 | 3 | 1 | 3 |
| Scaled identity | 4 | 2 | 8 | 16 | 4 |
| Zero diagonal matrix | 5 | 0 | 0 | 0 | 0 |
| Negative scaled identity | 2 | -1 | -2 | 1 | 2 |
Formula Used
For an order n matrix with diagonal value a, each entry is defined by:
Mij = a when i = j, and Mij = 0 when i ≠ j.
Trace = n × a
Determinant = an
Rank = n when a ≠ 0, otherwise 0.
When a = 1, the matrix becomes the standard identity matrix, which leaves any compatible vector or matrix unchanged after multiplication.
How to Use This Calculator
- Enter the matrix order you need, from 1 through 12.
- Set the diagonal value. Use 1 for the classic identity matrix.
- Optionally add a short matrix label and choose display precision.
- Press Submit to generate the matrix above the form.
- Review the trace, determinant, rank, and structural properties.
- Use Download CSV for spreadsheet work or Download PDF for print-ready output.
Frequently Asked Questions
1. What is an identity matrix?
An identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. It acts like the multiplicative identity for matrix operations.
2. Why must the matrix be square?
Identity matrices are defined only for square shapes because matrix multiplication needs matching row and column dimensions for the identity property to hold.
3. What happens if the diagonal value is not 1?
The result becomes a scaled identity matrix. It still stays diagonal and symmetric, but it no longer behaves as the standard identity unless the diagonal value equals one.
4. How is the determinant computed?
For a scaled identity matrix, the determinant equals the diagonal value raised to the matrix order. This follows from multiplying all diagonal entries together.
5. When is the matrix invertible?
The matrix is invertible whenever the diagonal value is not zero. If every diagonal entry is zero, the determinant becomes zero and no inverse exists.
6. Does the transpose change an identity matrix?
No. The transpose stays identical because all off-diagonal entries remain zero and the main diagonal does not move under transposition.
7. Can I use decimals or negative values?
Yes. The calculator accepts decimal and negative diagonal values, which helps when studying scaled identity matrices in linear algebra, modeling, and coding tasks.
8. What is the rank of the identity matrix?
A standard identity matrix of order n has rank n because every row and column is linearly independent. A zero diagonal version has rank zero.