Matrix transpose is one of 5 matrices formulas in the linear algebra section of this library, and it is used at high school · university level.
Why matrix transpose works
The operation reflects the whole array across the main diagonal, so the entry sitting in row i and column j moves to row j and column i. Diagonal entries stay where they are because they have matching indices. Nothing about the numbers changes; only their addresses are swapped.
What each symbol means
$i,j$ index row and column positions.
Matrix transpose: when it holds
For products, $(AB)^T=B^TA^T$; transposing twice returns the original matrix.
When it stops applying
For matrices with complex entries the plain transpose is rarely the tool you want, because it does not preserve lengths built from complex inner products. There the conjugate transpose, which also flips the sign of every imaginary part, takes its place.
Matrix transpose: a worked example
$\begin{bmatrix}1&2\\3&4\end{bmatrix}^T=\begin{bmatrix}1&3\\2&4\end{bmatrix}$.
The mistake to avoid
What people do: Students transpose a product term by term and write (AB)ᵀ as AᵀBᵀ.
Why it goes wrong: That version usually does not even make sense. If A is 2 by 3 and B is 3 by 2, then Aᵀ is 3 by 2 and Bᵀ is 2 by 3, so AᵀBᵀ is a 3 by 3 while the true answer is 2 by 2.
Do this instead: Reverse the order along with each transpose: (AB)ᵀ = BᵀAᵀ. Checking the dimensions is the fastest way to remember, since only the reversed order lines up.
Matrix transpose: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $i,j$ index row and column positions.
- Check the conditions before substituting. For products, $(AB)^T=B^TA^T$; transposing twice returns the original matrix.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most linear algebra slips.
Where this formula fits
- Subject
- Linear Algebra formulas — 17 entries in this library
- Topic
- Matrices
- Level
- High school · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where matrix transpose comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Matrix Operations — the lesson behind this formula: add, scale, and multiply matrices with attention to dimensions.
- Linear Algebra Calculator — check your substitution and the value it produces.
- Study linear algebra — the subject guide that explains the ideas these formulas compress.
- Linear Algebra Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 linear algebra formulas — the full grouped reference, or the complete formula library.
Questions about matrix transpose
What size is the transpose of a 2 by 5?
It is 5 by 2. The two dimensions simply trade places, which is why transposing is how you turn a row vector into a column vector and back again.
What makes a matrix symmetric?
A matrix is symmetric when it equals its own transpose, so the entry at row 1, column 3 matches the one at row 3, column 1. These matrices are special because their eigenvalues are always real.
Does transposing change the determinant?
No, det Aᵀ equals det A for every square matrix. That is why any rule about the rows of a determinant automatically holds for its columns as well.
Why does the order reverse in a product?
Because the entry in row i, column j of the product pairs row i of A with column j of B, and after transposing those become a column of Aᵀ and a row of Bᵀ. Only writing Bᵀ first puts them back into the right row-times-column arrangement.