## Introduction

Let us first understand what a matrix is. Oh, I am pretty sure you must be familiar with the matrix in Hollywood, but what does it mean in Mathematics? A matrix (plural: matrices) is a rectangular array or table of numbers(or symbols or expressions) arranged in rows and columns to represent a mathematical object or one of its properties. A matrix is usually denoted by a capital letter.

Below is an example of a matrix with three rows and three columns.

## Rank of a Matrix

The maximum number of linearly independent rows or columns of a matrix is called the rank of the matrix. Let us take matrix A. The rank of matrix A is denoted by ⍴(A). A row or column is linearly independent only if it cannot be represented as a scalar multiple of another row or column or a linear combination of other rows or columns.

If there is a Matrix A with m rows and n columns, the maximum rank of that matrix can be**: **m** **(if m<=n) or n** **(if n<m). The rank cannot exceed the number of rows and columns of the matrix.

Maximum Rank of a matrix A_{m*n }= ⍴(A) = minimum(m,n).

In layman language, the Rank of a Matrix is the number of unique rows (or columns) that are not made up of other rows(or columns).

When all the elements of a matrix become 0, it is said to be of rank 0. The rank of the null matrix is zero since it has no non-zero rows or columns.

### Properties

- The rank of a matrix of order m×n is ρ(A ) ≤ min{m, n} = minimum(m, n).
- A unit matrix of order m has a rank m.
- If a matrix A has order n×n and |A| ≠ 0, its rank will be n.
- If a matrix A has order n×n and |A| = 0, A's rank will be less than n.