Matrix Transpose Calculator
Enter the entries of a 2 × 2 or 3 × 3 matrix and this calculator instantly swaps its rows and columns to produce the transposed matrix Aᵀ, shown as a table.
Transpose a Matrix by Swapping Rows and Columns
Matrix Transpose Formula
Where:
- \(a_{ij}\) — the entry of matrix \(A\) in row \(i\) and column \(j\)
- \(A^{T}\) — the transpose of \(A\), whose entry in row \(i\), column \(j\) equals the entry of \(A\) in row \(j\), column \(i\)
- \(n\) — the number of rows and columns (2 or 3 for a square matrix)
The formula says that transposing a matrix means reflecting it over its main diagonal: every row of \(A\) becomes a column of \(A^{T}\), and every column becomes a row.
Matrix Transpose – Calculation Example
Example: transpose the 3 × 3 matrix with entries 1 through 9, arranged row by row.
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Swap rows and columns: the first column \((1, 4, 7)\) becomes the first row, the second column \((2, 5, 8)\) becomes the second row, and the third column \((3, 6, 9)\) becomes the third row.
\[ A^{T} = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix} \]
Result: the transposed matrix is \(A^{T} = \begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}\).
How to use the calculator
- In the Matrix size dropdown, choose 2 × 2 or 3 × 3 — the entry fields adjust automatically.
- Fill in every visible field of Matrix A (a₁₁, a₁₂, and so on) with the entries of your matrix, going row by row. The subscript tells you the position: a₁₂ is row 1, column 2.
- Read the result under Transposed matrix Aᵀ: it appears as a table with rows and columns swapped.
Leave all fields at their default values (a 3 × 3 matrix filled with 1 through 9) and the result will match the worked example above, so you can verify everything works as expected.
What is the transpose of a matrix?
The transpose of a matrix \(A\), written \(A^{T}\) (or sometimes \(A'\)), is obtained by turning every row into a column and every column into a row. Visually, it is a reflection of the matrix across its main diagonal — the diagonal running from the top-left corner to the bottom-right corner. Entries on the main diagonal stay exactly where they are; everything else moves to the mirrored position.
Although transposing looks like a purely mechanical operation, it carries real meaning. In data analysis, a table of measurements with subjects in rows and variables in columns becomes, after transposing, a table with variables in rows and subjects in columns. In computer graphics, transposes appear when converting between row-vector and column-vector conventions. In statistics, the transpose is a building block of the covariance matrix and of least-squares regression through expressions like \(A^{T}A\).
Interpreting the result is straightforward: compare the first row of the output with the first column of your input — they must be identical. If they are, the transpose is correct. For a square matrix, you can also check a useful property: the transpose of the transpose returns the original matrix, \((A^{T})^{T} = A\).
Two special cases are worth recognizing. If \(A^{T} = A\), the matrix is called symmetric — it reads the same across its diagonal. If \(A^{T} = -A\), it is skew-symmetric, which forces every diagonal entry to be zero. Try entering a symmetric matrix (for example, 1, 2, 3 in the first row and 2, 4, 5 in the second) and check that the output equals the input.
Properties, assumptions and tips
- Square matrices only. This calculator handles 2 × 2 and 3 × 3 matrices. The transpose itself is defined for any rectangular matrix — a 2 × 3 matrix transposes to a 3 × 2 matrix — but the entry fields here assume a square shape.
- Basic algebraic rules. Transposing respects addition, \((A + B)^{T} = A^{T} + B^{T}\), and reverses the order of multiplication, \((AB)^{T} = B^{T}A^{T}\). It also commutes with scalar multiplication: \((cA)^{T} = cA^{T}\).
- Any real numbers allowed. You can enter positive, negative, decimal, or zero values. Results are rounded to two decimal places for display, so values with more decimals may appear slightly shortened.
- Watch the sign convention. The transpose is not the same as the inverse, and it is not the conjugate transpose used for complex matrices (where signs of imaginary parts are also flipped — the matrix analogue of taking a complex conjugate).
If you are exploring related topics, a magic square is a fun symmetric-style matrix puzzle, and the elimination method for solving linear systems is another place where matrices and their entries are manipulated row by row.
Frequently Asked Questions
How do I transpose a matrix by hand?
Write the first row of your matrix as the first column of the new matrix, then the second row as the second column, and so on. Equivalently, each entry \(a_{ij}\) moves to position \(a_{ji}\), which is the mirror image across the main diagonal. For a 2 × 2 matrix only the two off-diagonal entries actually swap: \(\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{T} = \begin{bmatrix} a & c \\ b & d \end{bmatrix}\).
What is the difference between a transpose and an inverse?
The transpose simply swaps rows and columns — no arithmetic is done to the entries. The inverse \(A^{-1}\) is a different matrix that satisfies \(A A^{-1} = I\), and it only exists for square matrices with nonzero determinant. Computing an inverse requires division by the determinant, not just rearranging entries. A matrix can equal its own inverse (an involutory matrix), but that is a coincidence, not a rule.
Does transposing change the determinant or trace?
No. Both the determinant and the trace are unchanged by transposing: \(\det(A^{T}) = \det(A)\) and \(\operatorname{tr}(A^{T}) = \operatorname{tr}(A)\). This follows because the main diagonal entries stay in place. However, the eigenvalues of \(A^{T}\) also match those of \(A\), while the characteristic polynomial of related operations like the adjugate involves sign changes.
Can I transpose a non-square matrix, like 2 × 3?
Yes, the transpose exists for any rectangular matrix — a 2 × 3 matrix becomes a 3 × 2 matrix, with rows and columns trading roles. This calculator only provides entry fields for square 2 × 2 and 3 × 3 shapes. As a workaround for a 2 × 3 matrix, you can transpose it mentally by hand: write each of its two rows as a column and verify by reflecting over the diagonal.
Why does (AB)ᵀ = BᵀAᵀ and not AᵀBᵀ?
The order reverses because of how matrix dimensions must line up. If \(A\) is \(m \times n\) and \(B\) is \(n \times p\), then \(A^{T}\) is \(n \times m\) and \(B^{T}\) is \(p \times n\). The product \(A^{T}B^{T}\) is generally not even defined dimensionally, while \(B^{T}A^{T}\) is \(p \times m\), matching \((AB)^{T}\). This reversal is called the shoes-and-socks principle.
What is a symmetric matrix and how do I recognize one?
A matrix is symmetric when \(A^{T} = A\), meaning it is identical to its own reflection across the main diagonal — every off-diagonal pair satisfies \(a_{ij} = a_{ji}\). You can test this with the calculator: enter your matrix and check whether the output equals the input. Symmetric matrices have real eigenvalues and appear in covariance matrices and quadratic forms. A skew-symmetric matrix satisfies \(A^{T} = -A\) and must have zeros on its diagonal.
What common mistakes happen when transposing a matrix?
The most frequent error is swapping the wrong elements — people sometimes reflect across the anti-diagonal (from top-right to bottom-left) instead of the main diagonal, or they rotate the matrix 90° instead of reflecting it. Another mistake is assuming the transpose equals the inverse. Always verify: the first row of the output must exactly match the first column of the input, and diagonal entries must stay in place.