Zero matrix
A matrix is called a zero matrix if all its elements are zero. It is commonly represented as $$ 0_{mxn} = \begin{pmatrix} 0 & 0 & ... & 0 \\ 0 & 0 & ... & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & ... & 0 \end{pmatrix} $$ and may also be referred to as a null or empty matrix.
A zero matrix can be either square or rectangular, consisting of integers or real numbers, and it can have any number of rows and columns.
Example of a Zero Matrix
For instance, a 2x2 zero matrix has four elements, all set to zero:
$$ 0_{2x2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} $$
Here are a few other examples of null matrices:
$$ 0_{2x3} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} $$
$$ 0_{3x2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix} $$
$$ 0_{1x1} = \begin{pmatrix} 0 \end{pmatrix} $$
Properties of Null Matrices
1. Additive Identity
Null matrices act as the additive identity in matrix addition, meaning they leave the other matrix unchanged when added. The sum of any matrix \( M \) and a zero matrix of the same dimensions is simply \( M \):
$$ M + 0 = 0 + M = M $$
For example, consider the square matrix \( M \):
$$ M_{2x2} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
Adding \( M \) to a 2x2 zero matrix yields:
$$ M_{2x2} + 0_{2x2} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1+0 & 2+0 \\ 3+0 & 4+0 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
2. Multiplicative Absorber
The zero matrix also serves as an absorbing element in matrix multiplication. When a matrix is multiplied by a zero matrix of compatible dimensions, the product is always a zero matrix:
$$ M \cdot 0 = 0 \cdot M = 0 $$
For instance, consider the following square matrix \( M \):
$$ M_{2x2} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} $$
The product of \( M \) with a 2x2 zero matrix results in:
$$ M_{2x2} \cdot 0_{2x2} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \cdot \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 \cdot 0 + 2 \cdot 0 & 1 \cdot 0 + 2 \cdot 0 \\ 3 \cdot 0 + 4 \cdot 0 & 3 \cdot 0 + 4 \cdot 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} $$
