10. In an $n \times n$ matrix $\left( \begin{array} { c c c c c c c } 1 & 2 & 3 & \cdots & n - 2 & n - 1 & n \\ 2 & 3 & 4 & \cdots & n - 1 & n & 1 \\ 3 & 4 & 5 & \cdots & n & 1 & 2 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ n & 1 & 2 & \cdots & n - 3 & n - 2 & n - 1 \end{array} \right)$, let $a _ { i j }$ denote the entry in the $i$-th row and $j$-th column ($i , j = 1,2 \cdots , n$). When $n = 9$, $a _ { 11 } + a _ { 22 } + a _ { 33 } + \cdots + a _ { 99 } = $ $\_\_\_\_$ $45$. Analysis: $a _ { 11 } + a _ { 22 } + a _ { 33 } + \cdots + a _ { 99 } = 1 + 3 + 5 + 7 + 9 + 2 + 4 + 6 + 8 = 45$
(5 points) If $\cos \alpha = - \frac { 4 } { 5 }$ and $\alpha$ is an angle in the third quadrant, then $\sin \left( \alpha + \frac { \pi } { 4 } \right) = $
10. In an $n \times n$ matrix $\left( \begin{array} { c c c c c c c } 1 & 2 & 3 & \cdots & n - 2 & n - 1 & n \\ 2 & 3 & 4 & \cdots & n - 1 & n & 1 \\ 3 & 4 & 5 & \cdots & n & 1 & 2 \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ n & 1 & 2 & \cdots & n - 3 & n - 2 & n - 1 \end{array} \right)$,\\
let $a _ { i j }$ denote the entry in the $i$-th row and $j$-th column ($i , j = 1,2 \cdots , n$). When $n = 9$, $a _ { 11 } + a _ { 22 } + a _ { 33 } + \cdots + a _ { 99 } = $ $\_\_\_\_$ $45$.\\
Analysis: $a _ { 11 } + a _ { 22 } + a _ { 33 } + \cdots + a _ { 99 } = 1 + 3 + 5 + 7 + 9 + 2 + 4 + 6 + 8 = 45$