22.3. Elementary Row Operations

Try these first

1. 1.

   What is an example of row swapping on the following matrix: [[3, 5], [7, 2]]?

   Hint

   Consider the positions of the rows.

2. 2.

   If you multiply the row [2, 4] by 3, what do you get?

   Hint

   Multiply each element by 3.

3. 3.

   What are the three types of elementary row operations?

   • Row Addition
   • Row Multiplication
   • Row Substitution
   • Row Swapping
   • Row Addition
   • Scalar Multiplication
   • Row Swapping
   • Scalar Division
   • Row Elimination

   Hint

   Recall our discussions about manipulating matrices.

4. 4.

   True or False: Scalar multiplication can result in a row of zeros if the scalar is zero.

   • True
   • False

   Hint

   Think about the effects of multiplying by zero.

5. 5.

   Using the 3x3 matrix [[2, 3, 4], [1, 2, 1], [5, 6, 7]], perform a series of row operations to bring this matrix to REF and determine its rank.

   Hint

   Work systematically to isolate leading ones.

6. 6.

   Prove that using an elementary row operation will not change the rank of the matrix [[1, 0, 2], [2, 4, 6], [0, 0, 0]] by demonstrating a specific operation.

   Hint

   Examine how the relationships between rows stay intact.