Section 2.3 Matrix
A matrix is a rectangular array of numbers or symbols which describes various aspects of a phenomenon interrelated in some manner. If you use matrices to describe adjacency relations, then eigenvalues/vectors may mean one thing; if you use them to represent linear maps it means something else, etc. Matrices are a powerful tool in modern mathematics with a wide range of applications in subjects like sociology, demography, economics, statistics, and engineering, etc. The most significant contribution of matrices is their extensive use in the solution of a system of large number of simultaneous linear equations. The numbers in a matrix is arranged in rows and columns enclosed by a pair of brackets.
\begin{equation}
A={\begin{bmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{1n} \\
\cdots & \cdots & \cdots & \cdots\\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{bmatrix}} \tag{2.3.1}
\end{equation}
Here, the \(a_{ij}\) are called elements of \(i^{th}\) row and \(j^{th}\) column, they may be real or complex numbers or functions. The matrix A has \(m\) rows and \(n\) columns and is called a matrix of order \(m \times n\) (or, m by n). If \(m=n\text{,}\) then the matrix is called a square matrix. The main diagonal of a square matrix consists of the elements \(a_{11}, a_{22}, a_{33},\cdots, a_{mn}\text{.}\) The row matrix,
\begin{equation}
{\begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \end{bmatrix}}\tag{2.3.2}
\end{equation}
is called a row vector. The column matrix,
\begin{equation}
{\begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \\ \vdots \\ a_{m1} \end{bmatrix}}\tag{2.3.3}
\end{equation}
is called a column vector.
Two matrices (A and B) of the same order are said to be equal if and only if \(a_{ij} = b_{ij}\) for all \(i\) and \(j\text{,}\) e.g.,
\begin{equation}
A= {\begin{bmatrix} 2i\\1 \end{bmatrix}}; \quad B= {\begin{bmatrix} 2i\\1 \end{bmatrix}} \tag{2.3.4}
\end{equation}
If \(a_{ij}=0\) for all \(i\) and \(j\text{,}\) then \(A\) is called a null matrix, e.g.
\begin{equation}
A= {\begin{bmatrix} 0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\\ \end{bmatrix}} \tag{2.3.5}
\end{equation}
The multiplication of a matrix, \(A\) by a scalar, \(k\) is given by
\begin{equation}
kA=Ak\tag{2.3.6}
\end{equation}
where the elements of \(kA\) are \(ka_{ij}\) for all (\(\forall\)) \(i\) and \(j\text{,}\) e.g.,
\begin{equation}
A= 2{\begin{bmatrix} 1 & 2\\3 & 1\\ \end{bmatrix}} = {\begin{bmatrix} 2 & 4\\6 & 2\\ \end{bmatrix}}\tag{2.3.7}
\end{equation}
