Exercise 7

The operations that we perform in Gaussian elimination can be accomplished using matrix multiplication. This observation is the basis of an important technique that we will investigate in a subsequent chapter.

Let's consider the matrix $$A = \left[\begin{array}{rrr} 1 & 2 & -1 \\ 2 & 0 & 2 \\ -3 & 2 & 3 \\ \end{array}\right]$$.

1. Suppose that $$S = \left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$$. Verify that $SA$ is the matrix that results when the second row of $A$ is scaled by a factor of 7. What matrix $S$ would scale the third row by -3?

2. Suppose that $$P = \left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$$. Verify that $PA$ is the matrix that results from interchanging the first and second rows. What matrix $P$ would interchange the first and third rows?

3. Suppose that $$L_1 = \left[\begin{array}{rrr} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right]$$. Verify that $L_1A$ is the matrix that results from multiplying the first row of $A$ by $-2$ and adding it to the second row. What matrix $L_2$ would multiply the first row by 3 and add it to the third row?

4. When we performed Gaussian elimination, our first goal was to perform row operations that brought the matrix into a triangular form. For our matrix $A$, find the row operations needed to find a row equivalent matrix $U$ in triangular form. By expressing these row operations in terms of matrix multiplication, find a matrix $L$ such that $LA = U$.