21.5. Rank of a Matrix

Try these first

1. 1.

   What is the rank of a matrix with all zero rows?

   Hint

   Think about the definition of linearly independent vectors.

2. 2.

   If a matrix has 3 rows and 2 linearly independent rows, what is its rank?

   Hint

   Remember, rank corresponds to independent rows.

3. 3.

   What is the rank of a matrix?

   • Number of rows
   • Number of columns
   • Maximum number of linearly independent vectors

   Hint

   Focus on the meaning of independence in vectors.

4. 4.

   True or False: A matrix with more rows than columns can always have rank equal to the number of rows.

   • True
   • False

   Hint

   Consider the relationship of linear dependence.

5. 5.

   Given the matrix A = [[2, 4, 1], [6, 10, 3], [4, 8, 2]], find its rank by row-reduction and explain your steps.

   Hint

   Focus on identifying dependent rows.

6. 6.

   Demonstrate the impact of rank on the solutions of the linear system represented by A. If rank(A) < rank([A | b]), what can you conclude?

   Hint

   Think about the augmentation of the matrix and about linear equations.