Properties of Transpose of a Matrix

4.7 Properties of Transpose of a Matrix

1. $(A^T)^T=A$ for any matrix $A$.
2. $(kA)^T=kA^T$ for a constant $k$.
3. $(A+B)^T=A^T+B^T$, for $A,B$ of the same order.
4. $(AB)^T=B^TA^T$ — the transpose of a product reverses the order of the factors. More generally, $(A_1A_2\cdots A_n)^T=A_n^T\cdots A_2^TA_1^T$.
5. If $A$ is symmetric, $A^T=A$.
6. If $A$ is skew-symmetric, $A^T=-A$.
7. For any square matrix $A$: (a) $A+A^T$ is symmetric; (b) $A-A^T$ is skew-symmetric.

The symmetric/skew-symmetric decomposition

Combining property 7 with the scalar-multiplication property gives a standard and very useful decomposition: every square matrix $A$ can be written as the sum of a symmetric matrix and a skew-symmetric matrix,

Worked Examples

Example (Property 4). For matrices $A,B$ conformable for the product $AB$: compute $AB$ directly and transpose it, giving $(AB)^T$. Separately compute $A^T,B^T$ and the product $B^TA^T$ (note the reversed order — this is essential, since $A^TB^T$ would generally not even be conformable, or would give a different, wrong answer). Both routes give the same matrix, confirming $(AB)^T=B^TA^T$. …