How to calculate rank in matrix?
The maximum number of linearly independent rows in a matrix A is called the row rank of A, and the maximum number of linearly independent columns in A is called the column rank of A.
If A is an m by n matrix, that is, if A has m rows and n columns, then it is obvious that
Row rank of A ≤ m and Column rank of A ≤ n
For any matrix A,
the row rank of A = the column rank of A
Example:
Consider the matrix Y, shown below.
| Y = | | |||
| 1 | 2 | 3 | ||
| 2 | 3 | 5 | ||
| 3 | 4 | 7 | ||
| 4 | 5 | 9 |
Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer columns than rows, its maximum rank is equal to the maximum number of linearly independent columns.
Columns 1 and 2 are independent, because neither can be derived as a scalar multiple of the other. However, column 3 is linearly dependent on columns 1 and 2, because column 3 is equal to column 1 plus column 2. That leaves the matrix with a maximum of two linearly independent columns; e.g., column 1 and column 2. So the matrix rank is 2.