Matrix Transpose Calculator
Verified Calculation Engine
The online Matrix Transpose Calculator helps you calculate instantly and solve problems related to Matrices Determinants. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Transpose a matrix by swapping rows and columns. Enter Matrix A (m×n) to compute Aᵀ (n×m).
(Aᵀ)_ij = a_ji
The transpose of an m×n matrix A is the n×m matrix Aᵀ whose entries satisfy (Aᵀ)_ij = a_ji. Rows of A become columns of Aᵀ, and columns of A become rows of Aᵀ.
- Matrix A is a non-empty rectangular array of real numbers.
- All rows have the same number of columns.
- Floating-point display may round non-terminating decimals.
Inputs
Results
Worked Examples
Transpose a 2×2 square matrix
Square transpose: rows become columns; dimensions stay 2×2.
- matrix: [ [ 7, 8 ], [ 9, 10 ] ]
- result: [7,9], [8,10]
Transpose a 3×2 rectangular matrix
Rectangular transpose flips dimensions from 3×2 to 2×3.
- matrix: [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ]
- result: [1,3,5], [2,4,6]
Transpose a matrix with negatives and decimals
Negatives and decimals transpose the same way as integers.
- matrix: [ [ 1.5, -2.25 ], [ 0.5, 3.75 ], [ -4.5, 6.25 ] ]
- result: [1.5,0.5,-4.5], [-2.25,3.75,6.25]
About this calculator
Overview
The transpose of a matrix flips it over its main diagonal: rows become columns and columns become rows. If A is m×n, then Aᵀ is n×m and (Aᵀ)_ij = a_ji.
When to use
Use this matrix transpose calculator when you need Aᵀ for homework, changing orientation of data tables, preparing matrices for products such as AᵀA, or checking that (Aᵀ)ᵀ = A.
Inputs explained
- Matrix A — m×n numeric matrix to transpose
Formula / method
Read Matrix A with m rows and n columns. Build Aᵀ with n rows and m columns by placing each entry a_ji into position (i,j) of Aᵀ. Formula: (Aᵀ)_ij = a_ji. Square matrices keep the same size; rectangular matrices reverse dimensions (m×n → n×m).
Worked example
Example: A = [[1, 2], [3, 4], [5, 6]] is 3×2. Then Aᵀ = [[1, 3, 5], [2, 4, 6]] is 2×3, because the first column of A becomes the first row of Aᵀ.
Interpreting results
The Result Matrix is Aᵀ. Its dimensions are (columns of A) × (rows of A). Steps list each Aᵀ_ij = A_ji from your entered values.
Assumptions
- Matrix A is non-empty and rectangular (consistent row lengths).
- Entries are real numbers.
- Display precision may round intermediate floating-point values.
Limitations
- An empty or non-numeric matrix cannot be transposed — the calculator shows a validation message instead of a result.
- This tool performs a numeric transpose; it is not a symbolic computer algebra system.
- Critical engineering decisions should be cross-checked by hand or with certified software.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Read Matrix A with m rows and n columns. Build Aᵀ with n rows and m columns by placing each entry a_ji into position (i,j) of Aᵀ. Formula: (Aᵀ)_ij = a_ji. Square matrices keep the same size; rectangular matrices reverse dimensions (m×n → n×m).
Example Calculation
Example: A = [[1, 2], [3, 4], [5, 6]] is 3×2. Then Aᵀ = [[1, 3, 5], [2, 4, 6]] is 2×3, because the first column of A becomes the first row of Aᵀ.
Frequently Asked Questions
What is Matrix Transpose Calculator?
- The transpose of a matrix flips it over its main diagonal: rows become columns and columns become rows. If A is m×n, then Aᵀ is n×m and (Aᵀ)_ij = a_ji.
How does Matrix Transpose Calculator work?
- Read Matrix A with m rows and n columns. Build Aᵀ with n rows and m columns by placing each entry a_ji into position (i,j) of Aᵀ. Formula: (Aᵀ)_ij = a_ji. Square matrices keep the same size; rectangular matrices reverse dimensions (m×n → n×m).
Why use this Engineering Math calculator?
- Use this matrix transpose calculator when you need Aᵀ for homework, changing orientation of data tables, preparing matrices for products such as AᵀA, or checking that (Aᵀ)ᵀ = A.