# Square Matrices

So I'm studying a few special families of square matrices, the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices and I just had a few questions.

I know...

a diagonal matrix is if every nondiagonal entry is zero, $a_{ij}$=0 whenever $i$ doesn't equal $j$.

an upper triangular matrix is if all entries below the diagonal are zero, $a_{ij}=0$ whenever $i >j$.

a lower triangular matrix is if all entries above the diagonal are zero, $a_{ij}=0$ whenever $i < j$.

symmetric if $a_{ij}=a_{ji}$ for all $i$ and $j$.

But I was just wondering, can the diagonal matrices, upper triangular matrices, lower triangular matrices and symmetric matrices have the $0_{n\times n}$? Also I know the $I_{n\times n}$ matrix is in the diagonal matrices, but can it be in the other three types?

• Both the zero matrix and the identity matrix are diagonal, upper/lower triangular, and symmetric. Check the definitions. Commented Dec 9, 2016 at 4:55
• Just check the hypothesis Commented Dec 9, 2016 at 4:56

Take for example the definition of upper triangular matrix. It's a matrix which has $a_{ij}=0$ whenever $i>j$, but this doesn't mean that $a_{ij}\neq 0$ for $i\le j$, it's possible to have $a_{ij}=0$ if $i\le j$. Then both $O_{n\times n}$ and $I_{n\times n}$ are upper triangular matrices.