• Let m and n be positive integers. A rectangular array of numbers in $\mathbb{R}$of the form: $A= \begin{pmatrix}a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ddots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \ \end{pmatrix}$ is an m by n matrix, where m by n or $m \times n$ is the size of the matrix. such that m:  shows the horizontal rows and  n:  shows the vertical columns
• . Zero matrix: A matrix with all zero entries

$A = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}$

• Square matrix: A matrix in which the number of rows is equal to number of columns.

$A = \begin{pmatrix}
1 & 2 & 3\\
4 & 7 & 2 \\
8 & 1 & 0
\end{pmatrix}$ is a 3 by 3 matrix.

• Diagonal matrix: A square matrix with all zero entries except diagonal entries.

$A = \begin{pmatrix}
3 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 2
\end{pmatrix}$ is a 3 by 3 Diagonal matrix.

Lower triangular matrix: A square matrix whose entriesabove the main diagonal are all zero.

$A = \begin{pmatrix}
1 & 0 & 0 \\
2 &1 & 0\\
3 & 8 & 1
\end{pmatrix}$ is a 3 by 3 lower triangular matrix

• Two matrices $ A = (a_{i,j})_{mn}$ and $B = (b_{i,j})_{pq}$ are said to be equal if and only if: they have the same size i.e. $m = P$ and $n = q$ the corresponding entries are equal i.e. $a_{i,j} = b_{i,j}$

• Let $A = (a_{i,j})_{mn}$ be a matrix in the set of real numbers and $r \in \mathbb{R}$, then $rA = r (a_{i,j})_{mn} = (ra_{i,j})_{mn}$

•  Let A and B be matrices of real numbers of the same order,  then the following properties hold:r(A+B) = rA + rB ……. scalar multiplication is distributive over addition of matrices
• (r+s)A = rA + rB …….. scalar multiplications are distributive over matrix addition
• (rs)A = r(sA) = s(rA) and IA = AI = A

• Let $ A = (a_{i,j})_{mn}$ and $B = (b_{i,j})_{pq}$  be matrices, then the product AB exists If and only if: The number of column of A is equal to number of rows of B. i.e. n = p. order of AB is $m X q$.

$\text{Let}~A,~B~\text{and}~C~\text{be matrices in real numbers and}~r\in\mathbb{R},~\text{then}$

$\begin{aligned}
A(B+C)&=AB+AC\hspace{1cm}\text{…left distributive property}.\\
(A+B)C&=AC+BC\hspace{1cm}\text{…right distributive property}.\\
A(BC)&=(AB)C\hspace{1cm}\text{…associative property}.\\
r(AB)&=(rA)B=A(rB)\hspace{1cm}\text{…scalar multiplication.}
\end{aligned}$

Let A and B be matrices, A+B, AB are defined and $K \in \mathbb{R}$ then the following properties are satisfied:

      i   $ (A^{T})^{T} = A$                                       iv.   $(KA)^{T} = K A^{T}$

      ii .$(A + B)^{T} = A^{T} + B^{T}$                       v.   $I_{n}^{T} = I_{n}$  , where I is an identity matrix                 

iii . $(AB)^{T} = B^{T} A^{T}$