5.28 min read

Properties of the Transpose

The transpose interacts cleanly with all the matrix operations we have studied. Addition and scalar multiplication pass straight through: $(A + B)^t = A^t + B^t$ and $(rA)^t = rA^t$ . For products, the rule reverses order: $(AB)^t = B^t A^t$ .

When $A$ is invertible, so is $A^t$ , and its inverse is the transpose of $A^{-1}$ : $(A^t)^{-1} = (A^{-1})^t$ . This is sometimes written $A^{-t}$ . You can verify it directly: $A^t (A^{-1})^t = (A^{-1} A)^t = I^t = I$ .

A useful extension of the product rule: $(ABC)^t = C^t B^t A^t$ . Each factor transposes and the whole product reverses. This generalizes to any number of factors.

Formal View

Theorem 5.2 — Transpose Algebra

Let

A

and

B

m \times n

matrices,

r \in \mathbb{R}

, and let

A, B

have a well-defined product

AB

. Then:\begin{enumerate} \item

(A^t)^t = A

\item

(A + B)^t = A^t + B^t

\item

(rA)^t = rA^t

\item

(AB)^t = B^t A^t

\item If

A

is square and invertible:

(A^t)^{-1} = (A^{-1})^t =: A^{-t}

\end{enumerate}

Why This Matters

These algebraic rules make the transpose a well-behaved operation that fits seamlessly into calculations.

The rule $(AB)^t = B^t A^t$ is used constantly in deriving gradient formulas in machine learning
In numerical linear algebra, computing $(A^t A)^{-1} A^t$ (the pseudoinverse) relies on all these identities
When checking if a matrix is symmetric, the identity $(A + A^t)^t = A^t + A$ immediately shows $A + A^t$ is always symmetric

Learning Resources

Transpose properties

Khan Academy

Detailed walkthrough of the algebraic properties of matrix transpose.

8 min

Symmetric matrices and transpose

MIT OpenCourseWare

Gilbert Strang discusses symmetric matrices and the role of the transpose.

12 min

Quiz

Question 1

Which of the following is always symmetric (equals its own transpose) for any matrix $A$ ?

Question 2

If $A$ is invertible, then $(A^t)^{-1} = (A^{-1})^t$ .

Common Mistakes

Applying the transpose rule as $(AB)^t = A^t B^t$ — always reverse the order: $(AB)^t = B^t A^t$ .
Assuming every matrix plus its transpose equals the zero matrix — $A + A^t$ is symmetric, not zero.