Is the zero matrix upper and lower triangular as well as diagonal?

A zero square matrix is lower triangular, upper triangular, and also diagonal.

Provided it is a square matrix.

An upper triangular matrix is one in which all entries below the main diagonal are zero. Clearly this is satisfied.

A lower triangular matrix is one in which all entries above the main diagonal are zero. Clearly this is satisfied.

A diagonal matrix is one in which all non-diagonal entries are zero. Clearly this is also satisfied.

Hence, a zero square matrix is upper and lower triangular as well as a diagonal matrix.