22.4. Methods to Find Rank

Try these first

1. 1.

   What does the rank of a matrix indicate?

   Hint

   Think about dimensions and independence.

2. 2.

   Name one type of row operation.

   Hint

   Consider operations that change row positions.

3. 3.

   What is the rank of a zero matrix?

   • True
   • False

   Hint

   Consider the definition of rank and its parameters.

4. 4.

   To find the rank of a matrix, we count the number of non-zero rows in REF. Is this statement True or False?

   • True
   • False

   Hint

   Think about how we define row echelon form.

5. 5.

   Given the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 4 & 5 & 6 \end{bmatrix}$, apply both methods to find its rank.

   Hint

   Ensure to perform both calculations for confirmation.

6. 6.

   Can you find the rank of the matrix $\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{bmatrix}$? What does this reveal about linear dependence?

   Hint

   Remember to reduce the matrix carefully!