Chapter 2 - Matrices and Systems of Linear Equations - 2.1 Matrices: Definitions and Notation - Problems - Page 122: 32

See below

For $i > j$, $a_{ij}=0$ if $A$ is an upper-triangular matrix. If $A$ is a symmetric matrix as well, then $a_{ij} = 0 \forall i\ne j$. All elements above the main diagonal must therefore be zero. However, since that is a concept of a diagonal matrix, A must be one as well.