Calculator
Enter values, choose sizes, then transpose. Supports numbers and fractions like 7/4.
Formula Used
The transpose of a matrix swaps rows with columns. If matrix A has entries aij, then its transpose AT is defined by:
In practice: the first row becomes the first column, the second row becomes the second column, and so on.
How to Use This Calculator
- Select your matrix size using the Rows and Columns controls.
- Enter values in Matrix A, or load values from a CSV paste/upload.
- Choose display preferences like decimal places or fraction conversion.
- Press Transpose Matrix to see AT above the form.
- Use the download buttons to export the transposed matrix as CSV or PDF.
Example Data Table
A sample 3 × 3 matrix and its transpose.
| 2 | -1 | 5 |
| 0 | 3 | 4 |
| 7 | 1/2 | -6 |
| 2 | 0 | 7 |
| -1 | 3 | 1/2 |
| 5 | 4 | -6 |
What Transpose Delivers
A transpose converts an m × n matrix into an n × m matrix by swapping positions across the main diagonal. This calculator shows the new orientation immediately, which helps you verify structure before using the matrix in algebra, statistics, or programming workflows. When you transpose, every row becomes a column with the same ordered values, so your data layout is preserved while its direction changes. It also helps confirm symmetry, when A equals AT.
Input Rules and Validation
You can enter integers, decimals, negatives, or fractions like 3/8. Empty cells are treated as zero to keep the matrix complete. If a cell is not a valid number or fraction, it is highlighted so you can correct it. Size controls allow up to 20 rows and 20 columns, which is practical for most classroom and business calculations. The CSV paste and upload tools can populate the grid quickly, which is helpful for large tables.
Computation Details and Accuracy
The transpose operation follows the rule (AT)ij = Aji, so the element in row i, column j moves to row j, column i. Optional formatting keeps original input, or rounds to 0, 2, or 4 decimals for reporting. Fraction conversion can display decimal equivalents while preserving consistent output. Because the work is linear in the number of cells, results remain fast even near the maximum size.
Export Options for Reporting
After calculation, you can export the transposed matrix to CSV for spreadsheets and databases, using comma, semicolon, pipe, or tab delimiters. A lightweight PDF export produces a clean text table for quick sharing and printing. Ideal for audits and documentation. Exports match your display settings, so rounding and fraction conversion stay consistent across the page and downloaded files. CSV fields are safely quoted when needed, so separators inside values do not break the file.
Common Uses in Applied Maths
Transposes appear in dot products, covariance matrices, and least squares models, where orientation determines valid multiplication. They also support data cleaning by switching between “records as rows” and “features as rows.” In graphics and image processing, transposing can rotate data grids conceptually and simplify indexing in algorithms and simulations.
FAQs
1) What is the transpose of a matrix?
It is a new matrix formed by swapping rows and columns. Each element at position (i, j) moves to (j, i), changing the matrix size from m × n to n × m.
2) Can I enter fractions like 1/2?
Yes. Type fractions using the a/b format. You may keep them as fractions or enable conversion to decimals for a uniform numeric display and export.
3) Why are some cells highlighted as invalid?
Highlighting appears when a value is not a valid number or fraction, such as letters or broken formats. Fix the input and recalculate to generate a fully numeric transpose.
4) What matrix sizes are supported?
You can choose from 1 to 20 rows and 1 to 20 columns. This range covers typical academic problems and many practical datasets without cluttering the page.
5) Does rounding change the transpose operation?
No. Rounding only formats displayed values. The transpose rule still swaps indices exactly, and the shown numbers are simply rounded to your selected decimal places.
6) How do CSV and PDF downloads work?
Downloads use your latest calculated transpose stored in the session. CSV is ideal for spreadsheets, while the PDF provides a printable table with matrix sizes and values.