Rank of Matrix

Rank of Matrix
What is Rank of a Matrix?
Consider a matrix 𝐴 of order 𝑚 × 𝑛. It has 𝑚 rows and 𝑛 columns. 𝐴 can have square
submatrices of orders 1, 2, 3, … , 𝑙 only; 𝑙 being the least of 𝑚 and 𝑛. The rank of 𝐴 is the
largest number 𝑟, such that there exists a non-singular submatrix of order 𝑟. If all square
submatrices of 𝐴 are singular, the rank of 𝐴 is said to be 0.
Thus, any zero matrix has rank 0. And in fact, only zero matrices have rank 0. For a
nonzero matrix has at least one nonzero element and for the matrix, this element
happens to be a non-singular square submatrix of order 1, ensuring the rank of the
matrix to be at least 1.
Example 1
The matrix
2 1 0 3
𝐴 = (0 0 1 7)
3 0 0 4
has order 3 × 4.
It can have square submatrices of orders 1, 2 and 3 only.
𝐴 has (2) as a non-singular submatrix of order 1, for its determinant
|2|
=2
≠0
1 0
𝐴 has ( ) as a non-singular submatrix of order 2, for its determinant
0 1
1 0
| |
0 1
=1×1−0×0
=1−0
=1
≠0
2 1 0
𝐴 has (0 0 1) as a non-singular submatrix of order 2, for its determinant
3 0 0
2 1 0
|0 0 1|
3 0 0