Activity 3.1.4

This activity explores how the row operations of scaling, interchange, and replacement can be performed using matrix multiplication.

As an example, we consider the matrix $$A = \left[\begin{array}{rrr} 1 & 2 & 1 \\ 2 & 0 & -2 \\ -1 & 2 & -1 \\ \end{array}\right]$$ and apply a replacement operation that multiplies the first row by $-2$ and adds it to the second row. Rather than performing this operation in the usual way, we construct a new matrix by applying the desired replacement operation to the identity matrix. To illustrate, we begin with the identity matrix $$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$$ and form a new matrix by multiplying the first row by $-2$ and adding it to the second row to obtain $$R = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}.$$

1. Show that the product $RA$ is the result of applying the replacement operation to $A$.

2. Explain why $R$ is invertible and find its inverse $R^{-1}$.

3. Describe the relationship between $R$ and $R^{-1}$ and use the connection to replacement operations to explain why it holds.

4. Other row operations can be performed using a similar procedure. For instance, suppose we want to scale the second row of $A$ by $4$. Find a matrix $S$ so that $SA$ is the same as that obtained from the scaling operation. Why is $S$ invertible and what is $S^{-1}$?

5. Finally, suppose we want to interchange the first and third rows of $A$. Find a matrix $P$, usually called a permutation matrix that performs this operation. What is $P^{-1}$?

6. The original matrix $A$ is seen to be row equivalent to the upper triangular matrix $U$ by performing three replacement operations on $A$: $$A = \left[\begin{array}{rrr} 1 & 2 & 1 \\ 2 & 0 & -2 \\ -1 & 2 & -1 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & -4 & -4 \\ 0 & 0 & -4 \\ \end{array}\right] = U.$$ Find the matrices $L_1$, $L_2$, and $L_3$ that perform these row replacement operations so that $L_3L_2L_1 A = U$.

7. Explain why the matrix product $L_3L_2L_1$ is invertible and use this fact to write $A = LU$. What is the matrix $L$ that you find? Why do you think we denote it by $L$?