The product of any matrix by the scalar 0 is the null matrix. i.e., $0 \cdot A=0$. [where, $A$ is any matrix]

A matrix which is not a square matrix is called a rectangular matrix. For example a rectangular matrix is $A=\left[a_{i j}\right]_{m \times n}$, where $m \neq n$.

Matrix multiplication is distributive over addition.

e.g., For three matrices $A, B$ and $C$,

(i) $A(B+C)=A B+A C$

(ii) $(A+B) C=A C+B C$

If $A$ is a symmetric matrix, then $A^3$ is a symmetric matrix.

$$\begin{aligned} \because\quad A^{\prime} & =A \\ \therefore\quad \left(A^3\right)^{\prime} & =A^3 \\ & =A^3\quad \left[\because\left(A^{\prime}\right)^n=\left(A^n\right)^{\prime}\right] \end{aligned}$$

If $A$ is a skew-symmetric matrix, then $A^2$ is a symmetric matrix.

$$\begin{aligned} \because\quad A^{\prime} & =-A \\ \therefore\quad \left(A^2\right)^{\prime} & =\left(A^{\prime}\right)^2 \\ & =(-A)^2 \\ & =A^2 \end{aligned} \quad\left[\because A^{\prime}=-A\right]$$

So, A$^2$ is a symmetric matrix.