Question

What is a full rank matrix?

Answer 1

Hint: First, we will need to know the concept of matrix and its order, then we will discuss the full rank of the matrix.
A matrix is a rectangular entry with elements or variables.
Let us take the Square matrix, same order matrix-like $1 \times 1,2 \times 2,3 \times 3,.....n \times n$.
If an inverse exists, then the matrix is known as the non-singular because of the determinant non-zero.

Complete step-by-step solution:
The rank of the matrix is the number of the linearly independent rows or columns in the matrix, where the rank of the matrix is denoted as $\rho (A)$.
A matrix is said to be rank zero when all the elements become zero. The rank of the matrix is the dimension of the vector space obtained by its columns.
Rank cannot exceed more than the number of its order. We are only able to find the rank for the square matrix.
Hence Full rank matrix is nothing but the square matrix when its determinant is non-zero so that the inverse exists and it is known as the non-singular matrix (determinant non-zero)
Therefore the full rank matrix is a square matrix with determinant non-zero.
Additional information:
If for example take a $2 \times 2$ matrix which is $A = \left( {\begin{array}{*{20}{c}}
  2&1 \\
  4&3
\end{array}} \right)$and then find its determinant
If we see $\left| A \right| = \left| {\begin{array}{*{20}{c}}
  2&1 \\
  4&3
\end{array}} \right| = (2 \times 3) - (1 \times 4)$
 $ \Rightarrow 6 - 4$
$ \Rightarrow 2$
Hence, the determinant of the square matrix is non-zero and hence it is the Full Rank matrix.

Note: A matrix is nonsingular ($\left| A \right| \ne 0$) then we are able to find its inverse form.
If it singular ($\left| A \right| = 0$) then we cannot find its inverse form.
Also, in a matrix nonsingular matrix = invertible matrix.
If the order of the elements is not equal (not the same size), then it is called a non-square matrix. For example: the matrices of $1 \times 2,5 \times 7$