# Rank

Let *A* € M_{mxn}(**R**) , it is called **minor of order p** to the determinant of each submatrix pxp of A.

Then, the **rank of A** is the largest order of any non-zero minor in A.

Likewise, it is said that a row or column is** linearly dependent** if it is a linear combination of the others. Otherwise, it is called **linearly independent**. Then, the rank of a matrix is the size of the largest collection of linearly independent columns (or rows) of A.

*–Gauss*: we get the row echelon form of a matrix and the rank is the number of non-zero rows or columns.

Example 1:

*–Minors*. Example 1:

We look for non-zero determinant of order 1:

|1|≠ 0

Then a non-zero determinant of order 2, with the determinant of order one inside:

We edge the determinant of order 2 with the third row:

As all of them are zero, we remove the third row, because is linearly dependent:

We do the same with the fourth row

As all of them are zero, we remove the fourth row → rank(B) = 2

NOTE: in the beginning, we must remove the rows or columns of zeros, equal or proportional

Example 2:

We have a non-zero determinant of orders 1 and 2, and we edge this last one:

Then we edge this determinant with the fourth row:

**Exercises**:

1.- Calculate the rank of the matrix:

2.- Calculate the rank of the following matrices depending on the value of *a*:

Solutions: 1) rank(A) = 3

2) rk(B) = 1 if a = 2; rk(B) = 2 if a ≠ 2

rk(C) = 2 a € **R**

rk(D) = 2 if a = 1; rk(D) = 3 if a ≠ 1

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