关于“同阶对称矩阵不满足AB=BA则AB非对称”的证明正确性问询

Your proof is totally correct—great job breaking it down concisely! Let's unpack each step to reinforce why it works:

• First, we rely on the core transpose property for matrix multiplication: For any valid product of matrices (X) and (Y), ((XY)^t = Y^tXt). This is a foundational rule you can always count on for transposes.
• Since (A) and (B) are symmetric, by definition their transposes equal themselves: (A^t = A) and (B^t = B). Plugging these into the transpose property gives us ((AB)^t = B^tAt = BA).
• A matrix is symmetric if and only if it is equal to its own transpose. So if (AB \neq BA), then from the step above, ((AB)^t = BA \neq AB). This directly violates the definition of symmetry for (AB), so (AB) can't be symmetric.

As a side note, this is also equivalent to proving the contrapositive of the original statement: If (AB) is symmetric, then (AB = BA). Your direct proof is just as valid and efficient here—no need to overcomplicate it!

内容的提问来源于stack exchange，提问作者peterxd4