What is the rank for the matrix The question is from my line

What is the rank for the matrix?

The question is from my linear algebra class.

What is the rank for the matrix? [1 1 2 2 -1 3 5 - 1 8]. The question is from my linear algebra class.

Solution

The rank of a matrix  is defined as the dimension of the vector space spanned by its columns/ rows. Thus, in order to find the rank of the given matrix, we first transform the matrix to its reduced row echelon form (RREF) by the following row operations:

1.     add -2 times the 1st row to the 2nd row

2.     add -5 times the 1st row to the 3rd row

3.     multiply the 2nd row by -1/3

4.     add 6 times the 2nd row to the 3rd row

5.     add -1 times the 2nd row to the 1st row

Then the columns of the RREF of the given matrix are (1, 0, 0)^T, ( 0, 1, 0)^T and ( 5/3,1/3,0)^T. Apparently, the 3^rd column is a linear combination of the first two columns being equal to (5/3)C₁ + (1/3)C₂ where C₁ and C₂ are its 1^st and 2^nd columns respectively. Also, there are only 2 non-zero rows in the RREF of the given matrix. Therefore, the rank of the given matrix is 2.