Calculator Inputs
Formula Used
The identity matrix of size n is defined by entries: aᵢⱼ = 1 when i = j, and aᵢⱼ = 0 otherwise.
This tool also supports a scalar multiple kI, where: aᵢⱼ = k on the diagonal, and 0 elsewhere.
- trace(kI) = nk
- det(kI) = kⁿ
- rank(kI) = n if k ≠ 0, otherwise 0
- If k ≠ 0, then (kI)⁻¹ = (1/k)I
How to Use This Calculator
- Set the matrix size n for an n×n result.
- Choose scalar k to create I or kI.
- Select a view format to display the matrix clearly.
- Enable labels if you need indexed row and column names.
- Press Generate Matrix to see results above the form.
- Use the download buttons to export CSV or PDF.
Example Data Table
| Example | n | k | trace | det | rank | Matrix snapshot |
|---|---|---|---|---|---|---|
| Standard identity | 3 | 1 | 3 | 1 | 3 | [1 0 0; 0 1 0; 0 0 1] |
| Scaled identity | 4 | 2 | 8 | 16 | 4 | diag(2,2,2,2) |
| Zero diagonal | 5 | 0 | 0 | 0 | 0 | all zeros |
FAQs
1) What is an identity matrix?
It is a square matrix with ones on the diagonal and zeros elsewhere. It behaves like the number 1 for matrix multiplication.
2) What does kI mean?
kI is a scalar multiple of the identity matrix. Every diagonal entry becomes k, and all off-diagonal entries remain zero.
3) Why are trace and determinant shown?
They summarize key properties quickly. For kI, the trace is nk and the determinant is kⁿ, which helps verify the matrix and spot input mistakes.
4) When is the matrix invertible?
For kI, the matrix is invertible exactly when k is not zero. The inverse is (1/k)I, which is easy to compute and confirm.
5) Why does the PDF sometimes truncate?
The PDF is designed as a one-page preview for readability. For large n, it truncates lines, while the CSV always contains the full matrix.
6) Which CSV delimiter should I use?
Comma works for most spreadsheets. Semicolon helps in locales that use commas for decimals. Tab or pipe can reduce conflicts when importing into other tools.
7) Can I label rows and columns?
Yes. Choose numeric labels for simple indexing, or custom prefixes like r1 and c1 for clearer references in notes, reports, and homework steps.
8) How large can n be?
This page allows up to 60. Large matrices can be heavy to render in a table, so the preview may truncate, but downloads still generate the full matrix.