21.5.2. Methods to Find Rank

Try these first

1. 1.

   Define the rank of a matrix.

   Hint

   Think about linear independence.

2. 2.

   What is echelon form?

   Hint

   Visualize a staircase.

3. 3.

   What is the definition of rank?

   • A number of non-zero elements in a matrix
   • Maximum number of linearly independent rows or columns
   • The total number of rows and columns in a matrix

   Hint

   Focus on linear independence.

4. 4.

   True or False: Rank can only be determined using echelon form.

   • True
   • False

   Hint

   Think about different methods available to find rank.

5. 5.

   Given the $4 \times 4$ matrix $C = \begin{bmatrix} 1 & 0 & 0 & 5 \\ 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{bmatrix}$, find the rank after performing row reduction.

   Hint

   Focus on reducing the matrix to echelon form first.

6. 6.

   A $2 \times 3$ matrix $D$ has a rank of 2. If you added a third row that is a linear combination of the existing rows, what would the new rank be?

   Hint

   Consider the implications of linear combinations on independence.