Matrices - Concept, notation, order, equality, types of matrices, zero and identity matrix

A matrix is an ordered rectangular array of numbers or functions. These numbers or functions are called the elements or entries of the matrix. We denote matrices by capital letters such as $A, B, C$. Visually, a matrix is represented as a grid of values enclosed within square brackets $[ ]$ or parentheses $( )$, arranged in horizontal rows and vertical columns.

The order of a matrix indicates its dimensions. A matrix having $m$ rows and $n$ columns is said to be of order $m \times n$ (read as $m$ by $n$). If a matrix is of order $3 \times 2$, it visually appears as a tall rectangle with 3 horizontal layers and 2 vertical stacks, containing a total of $3 \times 2 = 6$ elements.

Two matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ are said to be equal if they are of the same order and each element of $A$ is equal to the corresponding element of $B$ ($a_{ij} = b_{ij}$ for all $i$ and $j$). Visually, equality means that if you overlay one matrix on top of the other, every single number in the grid matches its counterpart perfectly.

A Row Matrix is a matrix that has only one row, represented as $[a_{11} \, a_{12} \, ... \, a_{1n}]$ with order $1 \times n$. It looks like a single horizontal strip of numbers. A Column Matrix has only one column, represented as a vertical stack of numbers with order $m \times 1$.

A Square Matrix is a matrix where the number of rows is equal to the number of columns ($m = n$). In a square matrix, the elements $a_{11}, a_{22}, ..., a_{nn}$ are called the diagonal elements. Visually, these elements form a straight line from the top-left corner to the bottom-right corner, known as the principal diagonal.

A Diagonal Matrix is a square matrix where all non-diagonal elements are zero. A Scalar Matrix is a diagonal matrix where all diagonal elements are equal to a constant $k$. An Identity Matrix is a special scalar matrix where all diagonal elements are $1$ and all others are $0$, denoted by $I$. Visually, an identity matrix looks like a grid of zeros with a diagonal 'slash' of ones through the center.

A Zero Matrix or Null Matrix is a matrix in which all elements are zero. It is denoted by $O$. Unlike identity matrices, zero matrices do not have to be square and can be of any order $m \times n$. Visually, it is simply a rectangular block filled entirely with zeros.