We consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.

Let $\mathscr { A }$ be the set of all $3 \times 3$ symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
The number of matrices in $\mathscr { A }$ is
(A) 12
(B) 6
(C) 9
(D) 3

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if
(A) the first column of $M$ is the transpose of the second row of $M$
(B) the second row of $M$ is the transpose of the first column of $M$
(C) $M$ is a diagonal matrix with nonzero entries in the main diagonal
(D) the product of entries in the main diagonal of $M$ is not the square of an integer