The trace inner product on matrices

Let $n \geq 1$, and let $A$ be a matrix in $\mathcal{M}_n(\mathbb{R})$, $A = (a_{i,j})_{1 \leq i,j \leq n}$.

1. Calculate the trace $\text{tr}(A^t A)$ in terms of the $a_{i,j}$.

2. Show that the map $f$ defined on $\mathcal{M}_n(\mathbb{R}) \times \mathcal{M}_n(\mathbb{R})$ by $f(A,B) = \text{tr}(A^t B)$ is an inner product on $\mathcal{M}_n(\mathbb{R})$.

3. Show that for all symmetric matrices $A$ and $B$ in $\mathcal{M}_n(\mathbb{R})$,

$$(\text{tr}(AB))^2 \leq (\text{tr}(A^2))(\text{tr}(B^2))$$