Transpose of a Matrix - Definition, Properties, Example
Transpose of a Matrix - Definition, Properties, Example
TL;DR
The transpose of a matrix is the matrix you get by turning its rows into columns, written ATA^TAT — the entry in row iii, column jjj moves to row jjj, column iii. This article covers the definition, how the order flips, all the properties (including the reversal rule (AB)T=BTAT(AB)^T = B^T A^T(AB)T=BTAT), the link to symmetric matrices, and six worked examples.
What Is The Transpose Of A Matrix?
The transpose of a matrix AAA is a new matrix, written ATA^TAT (or sometimes A′A'A′), formed by interchanging the rows and columns of AAA. The entry that sits in row iii and column jjj of AAA lands in row jjj and column iii of ATA^TAT.
In symbols, if A=[aij]A = [a_{ij}] then AT=[aji]A^T = [a_{ji}].
Variable glossary. aija_{ij} is the entry in row iii, column jjj of AAA. The superscript TTT marks the transpose. The order of a matrix is its row-count by column-count, written m×nm \times nm×n.
Take a small example:
A=[251794],AT=[275914]
The first row of AAA, namely 2,5,1, becomes the first column of ATA^TAT. The second row becomes the second column. Nothing is added or removed; the numbers are simply read in the other direction.
How Does The Order Change When You Transpose?
If AAA has order m×nm \times n, then ATA^TAT has order n×mn \times m. The dimensions swap.
A 2×32 \times 3 matrix transposes to a 3×22 \times 2 matrix. A 4×14 \times 1 column vector transposes to a 1×41 \times 4 row vector. Only a square matrix keeps the same order after transposing, because n×nn \times n stays n×nn \times n.
This order-swap is the detail that controls everything later. It is exactly why the transpose of a product reverses — the orders have to stay compatible for multiplication of matrices to remain defined.
What Are The Properties Of The Transpose?
Five properties carry almost every transpose problem you will meet. Each one has a short reason behind it, not just a rule to memorize.
Transpose of a transpose returns the original: (AT)T=A.
Transpose distributes over addition: (A+B)T=AT+BT.
Scalars pass straight through: (kA)T=kAT.
Transpose of a product reverses the order: (AB)T=BTAT.
Determinant is unchanged: det(AT)=det(A).
The reversal rule generalizes to longer products: (ABC)T=CTBTAT.
How Does The Transpose Define Symmetric Matrices?
A square matrix is symmetric when it equals its own transpose: A=ATA = A^TA=AT. The entries mirror across the main diagonal — aij=ajia_{ij} = a_{ji} for every pair. A square matrix is skew-symmetric when AT=−AA^T = -A.
Examples Of The Transpose Of A Matrix
Example 1
Find the transpose of A=[3816].
Read each row of AAA as a column.
AT=[3186].
Example 2
Transpose the rectangular matrix B=[2−10453].
BT=[24−1503].
Example 3
Find (AB)T for A=[1201] and B=[3045].
Example 4
Verify (AT)T=A.
Example 5
Is A=[2997] symmetric?
Example 6
Confirm the scalar rule (3A)T=3AT.
Why The Transpose Matters: "The bridge between rows and columns"
The transpose exists because rows and columns are not interchangeable inside a matrix, yet many real computations need to treat them as if they were. The transpose is the formal bridge that lets you cross between the two.
Where that bridge does real work:
Least-squares and regression.
Symmetry detection.
How computers store grids.
Key Takeaways
The transpose of a matrix swaps its rows and columns.
The core properties include:
- (AT)T=A, (A+B)T=AT+BT, (kA)T=kAT.
- The transpose of a product reverses the factor order: (AB)T=BTAT.
- A matrix is symmetric when A=ATA, skew-symmetric when AT=−A.
A Practical Next Step
Work through the provided examples to lock in the operation.