Chapter 2 - Matrices and Systems of Linear Equations - 2.1 Matrices: Definitions and Notation - Problems - Page 122: 25

$\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{bmatrix}$

Make $a_{ij}=0$, if $i\lt j$. That is, $a_{12}=a_{13}=a_{23}=0$ Also, $a_{ji}=-a_{ij}$. So: $a_{21}=a_{12}=0$ $a_{31}=a_{13}=0$ $a_{32}=a_{23}=0$ The elements, $a_{ii}$, of the main diagonal must be equal to zero because $a_{ii}=-a_{ii}$ The only possibility is: $\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{bmatrix}$