Let $A$ be a symmetric and PD matrix. Prove that the matrix $B=A_2-\ell_1u_1^T$ which you get in the LU decomposition process is symmetric. I understand that $A_2$ is the sub-matrix of $A$ (from $(2,2)$), $\ell_1$ is the first column of the $L$ matrix and $u_1$ is the first row of the $U$ matrix.
I also know that I need to prove that $B=B^T$. Matrix $A$ is symmetric so $A=A^T$ and it's also PD so for every $x\neq 0$ we have $x^TAx>0$. But should I properly prove that $B$ is symmeric?
Firstly, $\ell_1u_1^T$ is not the same size as $A_2$, unless you take the matrices $L_2$ and $U_2$. In this case, we don't have that $B$ is symmetric, which you can easily verify by checking an arbitrary $L$ and $U$. If this is not what you meant, perhaps post a photo of the question, from wherever you pulled it from?