Matrix Transpose Calculator
Turn the rows of a real-number matrix into columns instantly, with a live result and a validated Excel workbook you can keep.
Matrix A
Choose a size from 1 to 4 rows and columns, then enter one finite real number in each cell.
Transposed matrix
Each element A(i,j) moves to Aᵀ(j,i); numerical values themselves do not change.
How to use the matrix transpose calculator
What this calculator does
This calculator takes a rectangular matrix of real numbers and forms its transpose, written Aᵀ. Transposition does not add, multiply, round, or otherwise alter an entry. It only swaps the two position indices: the value stored at row i, column j moves to row j, column i. The result is therefore exact for every accepted finite input. The tool is designed for ordinary real-number matrices from 1 × 1 through 4 × 4; it does not calculate an inverse, determinant, product, conjugate transpose, or any other matrix operation.
When to use it
Use it when checking linear-algebra homework, reorienting a small data table from observations-by-variables to variables-by-observations, preparing a row/column arrangement for another matrix operation, or verifying that a hand-written transpose has the right dimensions and element positions. If you are implementing the same operation in software, the official NumPy transpose documentation shows how the standard two-dimensional transpose extends to arrays with more axes.
How to calculate
- The calculator opens with a complete demonstration matrix, so the transpose and the example XLSX are available immediately. Review that example first if you want to see the row-to-column pattern.
- Set Number of rows and Number of columns. The matrix-entry area resizes immediately. Existing values are preserved where their row and column still exist.
- Fill every Matrix entries cell. Results update live; there is no separate Calculate button because transposition is immediate once the matrix is complete.
- Read Transpose size, Original dimensions, Element count, the shape pills, the row/column mapping summary, and the displayed Aᵀ matrix. Use Download Excel to save the current validated model. Reset clears the demonstration values, returns the size selectors to a neutral 2 × 2 state, clears results, and disables export until every matrix cell contains a valid number again.
Input guide
Number of rows is a required integer selector from 1 to 4. For example, choose 3 for a matrix with three horizontal rows. Raising this value adds an original row and therefore adds a column to Aᵀ. A common mistake is to enter the desired transpose size rather than the original matrix size. Number of columns is also a required integer selector from 1 to 4. Choosing 2 means every original row has two entries; in the transpose, that becomes two rows. Again, specify the shape of A, not Aᵀ.
Matrix entries are required finite real numbers. The accepted format is a plain base-10 integer or decimal with an optional leading plus or minus sign, such as -2, 0, 3.5, or .25. Thousands separators, decimal commas, fractions such as 1/2, and scientific notation such as 1e3 are intentionally rejected so pasted text cannot be silently reinterpreted. Changing any entry changes exactly one corresponding location in Aᵀ and leaves all other values untouched.
Output guide
Transpose size is the dimensions of Aᵀ and is always the reverse of the original dimensions. Original dimensions repeats the shape of A for comparison. Element count is rows × columns; transposition never changes that count. The header pills Original, Transpose, and Elements are compact versions of the same canonical values. The mapping summary states how original rows become transpose columns. The displayed Aᵀ matrix is the exact positional result: a zero remains zero, negative values keep their signs, and high or low values carry no special meaning because transposition changes position rather than magnitude.
Worked example
The startup example uses a 3 × 2 matrix with rows [3, – 1], [0, 2], and [1, – 1]. The first row becomes the first column, the second row becomes the second column, and the third row becomes the third column. So Aᵀ has two rows: [3, 0, 1] and [ – 1, 2, – 1]. Its displayed size is exactly 2 × 3 and its element count remains 6. This is the same result written to the initial validated workbook.
Why transposition works
A matrix is indexed by two coordinates. If A(i,j) means “the entry in row i and column j,” transposition defines a new matrix by reversing those coordinates: Aᵀ(j,i) = A(i,j). That one rule explains both the value placement and the dimension swap. An m × n matrix has m rows and n columns, so after the row index becomes the column index, Aᵀ must have n rows and m columns. The Wolfram MathWorld entry on transpose gives the standard indexed definition and notation.
For a square matrix the dimensions happen to remain the same, but the off-diagonal entries still exchange positions. A symmetric matrix is the special case in which that swap reproduces the original matrix exactly. Rectangular matrices make the dimensional change more obvious: 2 × 4 becomes 4 × 2, while 1 × 4 becomes 4 × 1.
Useful transpose properties and common mistakes
Applying the operation twice returns the starting matrix: (Aᵀ)ᵀ = A. Transposition also distributes over addition, and the transpose of a product reverses factor order: (AB)ᵀ = BᵀAᵀ. Those identities are useful when checking longer linear-algebra derivations, but this calculator intentionally performs only the single-matrix transpose shown on screen.
The most common practical errors are swapping the size before entering the original matrix, reading across a column instead of down it, or changing a sign while copying a value. A transpose does not negate values. It also is not the same as an inverse: A⁻¹, when it exists, is defined by matrix multiplication with A to produce an identity matrix, whereas Aᵀ always exists for any finite rectangular matrix. When checking a result by hand, confirm the dimensions first, then pick two or three entries and verify that A(i,j) appears at Aᵀ(j,i).