Algebra - Matrices: Concept, Types, Transpose, Symmetric and Skew-symmetric matrices

Definition and Order: A matrix is an ordered rectangular array of numbers or functions. The numbers are called elements or entries. A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$. Visually, the order represents the 'dimensions' of the grid, where $m$ is the vertical count and $n$ is the horizontal count.

Types of Matrices: A Row Matrix has only one horizontal row $[a_{11} a_{12} ... a_{1n}]$, while a Column Matrix has only one vertical column. A Square Matrix has an equal number of rows and columns ($m = n$). A Zero or Null Matrix is one where every element is $0$, visually appearing as an empty grid of zeros.

Diagonal and Identity Matrices: A Diagonal Matrix is a square matrix where all non-diagonal elements are zero. An Identity Matrix ($I$) is a diagonal matrix where all diagonal elements are $1$. Visually, the Identity matrix shows a 'strip' of ones running from the top-left corner to the bottom-right corner, with all other spaces filled by zeros.

Transpose of a Matrix: The transpose of a matrix $A$, denoted by $A^T$ or $A'$, is formed by interchanging its rows and columns. If $A$ is of order $m \times n$, then $A^T$ is of order $n \times m$. Visually, this is like rotating the matrix over its main diagonal axis so that the first row becomes the first column.

Symmetric Matrix: A square matrix $A$ is said to be symmetric if $A^T = A$. This means the element at $a_{ij}$ is equal to the element at $a_{ji}$. Visually, if you fold the matrix along its main diagonal, the elements on both sides match perfectly like a mirror image.

Skew-Symmetric Matrix: A square matrix $A$ is said to be skew-symmetric if $A^T = -A$. This implies $a_{ij} = -a_{ji}$ for all $i, j$. Crucially, all diagonal elements of a skew-symmetric matrix must be zero ($a_{ii} = 0$). Visually, the diagonal is a line of zeros, and the elements across the diagonal are identical in magnitude but opposite in sign.