TITLE: Is the zero matrix upper and lower triangular as well as diagonal?
QUESTION [11 upvotes]: From what I can tell from the definitions of a lower-triangular, upper-triangular, and diagonal matrices, I've come to the conclusion that the zero matrix is in the set of all of each type of matrix.  Is this correct?

REPLY [4 votes]: 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.