Matrix Transpose Calculator
This matrix transpose calculator finds the transpose of a matrix — square or rectangular, 2×2 up to 4×4 — with steps, showing where every single entry lands rather than just handing you the finished grid. For a square matrix it also names the type — symmetric, skew-symmetric or neither — and splits it into one of each.
Start with a rectangle, because this is the one operation that welcomes them. The 2×3 matrix 1 2 3 / 4 5 6 transposes to the 3×2 matrix 1 4 / 2 5 / 3 6. The first row has become the first column, the second row the second column, and the order has reversed from 2×3 to 3×2. No arithmetic happened at all: every number is one you typed, standing somewhere else.
How to Transpose a Matrix
The entry in row i and column j moves to row j and column i. Written the usual way, the transpose of a matrix A is Aᵀ or AT, and the rule is (Aᵀ)ij = Aji. Rows become columns, columns become rows, and the dimensions swap: an m × n matrix comes back as n × m. So a 2×3 transposes to a 3×2, a 3×2 back to a 2×3, and a 3×3 stays 3×3.
The operation is also called transposition, and it is younger than you might expect: Arthur Cayley introduced it in 1858, long after determinants were in common use.
Geometrically it is a reflection in the main diagonal — the one running from the top-left corner downwards, also called the principal diagonal. That is worth holding on to, because it explains the two things that follow. Entries sitting on that diagonal are their own reflection, so they never move. Everything else moves in pairs: whatever is at (1, 2) trades places with whatever is at (2, 1). A 4×4 matrix looks like sixteen separate journeys but is really four fixed entries and six swaps, which is how this calculator lays it out.
Symmetric and Skew-Symmetric Matrices
Once you can see the reflection, two special cases fall out immediately, and both apply only to a square matrix — a rectangular one changes shape when transposed, so it can never equal itself.
A symmetric matrix satisfies Aᵀ = A: every entry equals its mirror image, so each swap pair holds two identical numbers. 1 1 −1 / 1 2 0 / −1 0 5 is one. Covariance matrices, distance tables and the adjacency matrix of an undirected graph all come out this way without anyone arranging it.
A skew-symmetric matrix satisfies Aᵀ = −A: every entry is minus its mirror image, as in 0 2 −3 / −2 0 4 / 3 −4 0. Notice the diagonal. A diagonal entry is its own mirror image, so it would have to equal its own negative, and zero is the only number that does — which is why the main diagonal of a skew-symmetric matrix is always zeros. That also makes the zero matrix the single matrix that is symmetric and skew-symmetric at the same time, a favourite examiner's question that this calculator answers by name.
Splitting a Matrix into Symmetric and Skew-Symmetric Parts
Most square matrices are neither, and that is not a dead end. Every square matrix splits into a symmetric part plus a skew-symmetric part, in exactly one way:
A = ½(A + Aᵀ) + ½(A − Aᵀ)
The first bracket is always symmetric, the second always skew-symmetric, and the two always add back to A. Take 1 2 3 / 4 5 6 / 7 8 9, whose transpose is 1 4 7 / 2 5 8 / 3 6 9. Half the sum gives the symmetric part 1 3 5 / 3 5 7 / 5 7 9, and half the difference gives the skew-symmetric part 0 −1 −2 / 1 0 −1 / 2 1 0. Add those two entry by entry and the original matrix comes straight back.
This is the question exam papers phrase as “express A as the sum of a symmetric and a skew-symmetric matrix”, and it is the part most calculators leave out — they will hand you Aᵀ and stop. This one works both halves, checks that the first really equals its own transpose and the second really equals minus its own, and adds them back to A before showing you anything.
Properties of the Transpose
Four rules cover almost everything you will be asked to prove, and each one connects this operation to another.
(Aᵀ)ᵀ = A. Transposing twice returns the original, because reflecting twice in the same line puts everything back.
(A + B)ᵀ = Aᵀ + Bᵀ. Transposing distributes over a sum, and over a scalar too: (kA)ᵀ = kAᵀ. You can check that against the matrix addition and subtraction calculator.
(AB)ᵀ = BᵀAᵀ. Transposing a product reverses the order of the factors, which surprises people the first time and is the most examined of the four. If you need the transpose and the multiplication together, work out AB on the matrix multiplication calculator, bring the answer back here, and compare it with BᵀAᵀ done the other way round.
det(Aᵀ) = det(A) and (A⁻¹)ᵀ = (Aᵀ)⁻¹. Transposing leaves the determinant untouched, and it commutes with taking the inverse — transpose then invert, or invert then transpose, and you land in the same place.
How to Use This Matrix Transpose Calculator
The screen has three parts: the answer band at the top, the matrix you fill in below it, and the keypad underneath.
1. Set the size. Two rows of number buttons sit above the grid, one for rows and one for columns, from 1 to 4. Any shape is fine here — a 1×4 row becomes a 4×1 column, and nothing is turned away.
2. Fill in the cells. Tap a cell to select it, then use the keypad. A cell takes a whole number, a decimal such as 0.25, or a fraction such as 3/4, and anything left blank counts as zero. Fractions stay fractions right through to the answer.
3. Press =. Aᵀ appears at the top with its new dimensions stated. If the matrix is square, the caption also tells you whether it is SYMMETRIC or SKEW-SYMMETRIC.
4. Open the moves. This is the part that makes it a transpose calculator with steps rather than an answer box. The button marked Show every move lays it out: for a rectangle, one block per entry showing exactly where it lands; for a square, one block for the diagonal that stays put and one for each swap pair, then the type check, the split, and the proof. A 3×3 comes to seven blocks and a 4×4 to ten.
Every key and control, in one place:
| rows / columns | Sets the size, 1 to 4, each one on its own; rectangles are welcome |
| the cells | Tap to select; blank counts as zero; fractions and decimals both accepted |
| ← → | Move to the previous or next cell, wrapping round the whole grid |
| ↑ ↓ | Move up or down a row, for filling a column at a time |
| ± | Flips the sign of the cell you are in, so a minus cannot be typed twice by accident |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way to the answer |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the cell you are in |
| AC | Clears the whole matrix and starts again |
| = | Transposes it |
| My keyboard | Switches that cell to your own keyboard, for anything the pad does not carry |
| copy | The icon in the band copies Aᵀ, row by row |
Change anything after an answer is showing and the band dims and asks you to press = again, so what you are reading always belongs to what is in the cells. There are no limits, no ads and no account.
Matrix Transpose FAQ
How do you find the transpose of a matrix?
What happens to the order of a matrix when it is transposed?
What is a symmetric matrix?
What is a skew-symmetric matrix and why is its diagonal zero?
How do you express a matrix as the sum of a symmetric and a skew-symmetric matrix?
Does this transpose calculator show the steps?
Can a rectangular matrix be transposed?
Rows into columns, the diagonal staying put, and everything else trading places in pairs. Once the reflection is clear, symmetric, skew-symmetric and the split into both stop being definitions to memorise and start being things you can see.