Question

Is a zero matrix symmetric?

Answer 1

First of all, let us know, what is a zero matrix? So, a zero matrix is a matrix with all its elements as $0$. Next we have to know, what is a symmetric matrix? Therefore, a symmetric matrix is a square matrix such as $A = \left( {{a_{ij}}} \right)$ in which, the terms opposite to its main diagonal are equal, that is, ${a_{ij}} = {a_{ji}}$ for all $i$ and $j$, also we can say that, the transpose of the matrix is equal to the original matrix, that is, $A = {A^T}$. So, in this case, we have to consider a zero square matrix, that is, a matrix with equal number of rows and columns, and with all its elements as zero.

Complete step by step answer:
So, if we consider the conditions of a symmetric matrix, than the elements opposite to the main diagonal have to be equal.So, in a zero matrix, the elements opposite to the main diagonal are always equal, every element being $0$, that is, ${a_{ij}} = {a_{ji}} = 0$, for all $i$ and $j$.Also, the same condition can be said in another way, like, $A = {A^T}$, for the matrix to be symmetric.An example of symmetric matrix is as follows:
\left( {\begin{array}{*{20}{c}} x&a;&b; \\\ a&y;&c; \\\ b&c;&z; \end{array}} \right)
Notice that the transpose of this matrix will be equal to: \left( {\begin{array}{*{20}{c}} x&a;&b; \\\ a&y;&c; \\\ b&c;&z; \end{array}} \right)
Hence, the matrix is equal to its transpose itself. Hence, the matrix is a symmetric matrix.
Now, zero matrix is equal to
\left( {\begin{array}{*{20}{c}} 0&0&0 \\\ 0&0&0 \\\ 0&0&0 \end{array}} \right)

The transpose of this zero matrix is as follows:
\left( {\begin{array}{*{20}{c}} 0&0&0 \\\ 0&0&0 \\\ 0&0&0 \end{array}} \right)
Now, the transpose of the matrix is equal to the matrix itself. So, we can conclude that the zero matrix is a symmetric matrix.So, we can say, for a zero matrix also, $A = {A^T}$, for the elements to be all $0$.

Therefore, we can conclude that the zero matrix is a symmetric matrix.

Note: Interestingly, the zero matrix is also a skew-symmetric matrix, as, in the skew symmetric matrix the opposite terms of the main diagonal are negations of each other, that is, ${a_{ij}} = - {a_{ji}}$, for all $i$ and $j$. So, in a zero matrix, the elements are also negations of each other about the main diagonal as all elements are equal to $0$. So, we can also say that a zero matrix is also a skew-symmetric matrix.