Exercise 511Linear algebra › Matrices and linear systemsL1Added 3 Nov 2026

Properties of the transpose

#transpose#properties#matrix operations

Statement

Let $A, B$ be matrices of compatible sizes and $\lambda \in \mathbb{R}$ . Prove that
$(A+B)^T = A^T + B^T, \qquad (\lambda A)^T = \lambda A^T, \qquad (AB)^T = B^T A^T.$

Hint

For the third identity, compare the entry $(i,j)$ on both sides using the definition of matrix product.

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Solution

First identity. The entry $(i,j)$ of $(A+B)^T$ is the entry $(j,i)$ of $A+B$ , which is $a_{ji}+b_{ji}$ . This is exactly the sum of the entries $(i,j)$ of $A^T$ and $B^T$ .

Second identity. The entry $(i,j)$ of $(\lambda A)^T$ is $\lambda a_{ji} = \lambda (A^T)_{ij}$ .

Third identity. The entry $(i,j)$ of $(AB)^T$ is the entry $(j,i)$ of $AB$ :
$(AB)_{ji} = \sum_k a_{jk} b_{ki}.$
The entry $(i,j)$ of $B^T A^T$ is
$\sum_k (B^T)_{ik}(A^T)_{kj} = \sum_k b_{ki} a_{jk}.$
The two sums are equal (same terms, reordered), so $(AB)^T = B^T A^T$ .

Properties of the transpose

Statement

Solution

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