Question

Are all zero matrices equal?

Answer 1

Hint: A zero matrix is a matrix which has all its elements as 0. Now we can say two matrices are equal if all the entries of the matrix are equal. Now we will use this knowledge to check if all the zero matrices are equal.

Complete step by step answer:
Now let us first understand the concept of matrices.
A matrix is nothing but a rectangular array of terms consisting of rows and columns.
Suppose we have a matrix with m rows and n columns then the matrix has order $m\times n$ .
Now let us try to write a matrix of order $2\times 3$ . Now the matrix will have 2 rows and 3 columns.
$\left[ \begin{align}
  & \begin{matrix}
   1 & 1 & 1 \\
\end{matrix} \\
 & \begin{matrix}
   1 & 1 & 1 \\
\end{matrix} \\
\end{align} \right]$ .
Now let us understand the concept of zero Matrix. A matrix is said to be a zero matrix if all its entries are 0.
Hence we can say that $\left[ \begin{align}
  & \begin{matrix}
   0 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 0 \\
\end{matrix} \\
\end{align} \right]$ is a zero Matrix.
Now two matrices are said to be the same if all the entries are the same. But we can compare only the matrix with the same order.
Hence let us say we have a matrix of order $2\times 3$ and another matrix of order $3\times 2$ then we cannot compare the two matrices.
Now if we have a zero matrix of the same order then definitely we can say that the two matrices are equal. But if we have two zero matrices of different order then the matrices are not equal.
For example consider $\left[ \begin{align}
  & \begin{matrix}
   0 & 0 & 0 \\
\end{matrix} \\
 & \begin{matrix}
   0 & 0 & 0 \\
\end{matrix} \\
\end{align} \right]$ and $\left[ \begin{matrix}
   0 & 0 \\
\end{matrix} \right]$ are both zero matrices but not equal.

Note:
Now note that even for addition and subtraction the matrices need to be of the same order. Though multiplication of matrices is only possible if the order of matrices are of the form $m\times n$ and $n\times r$. Also note that division of matrices is not defined.