Definition of 0 matrix

Definition: The zero matrix (or null matrix), denoted by $0$ or $0_{m \times n}$, is the $m \times n$ matrix in which every entry is zero:

$$0_{m \times n} = \begin{pmatrix} 0 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \end{pmatrix}$$

The zero matrix serves as the additive identity in the vector space of matrices: $A + 0 = A$ for any matrix $A$ of the same size.

Properties:

• $A + 0 = A$ and $0 + A = 0$
• $A - A = 0$
• $0 \cdot A = 0$ and $A \cdot 0 = 0$
• $\text{rank}(0) = 0$
• $\det(0_{n \times n}) = 0$