Question 8.1.2: Performing Matrix Row Operations Use the matrix [3 18 -12 21...
Performing Matrix Row Operations
Use the matrix
\left[\begin{array}{rrr|r}3 & 18 & -12 & 21 \\1 & 2 & -3 & 5 \\-2 & -3 & 4 & -6\end{array}\right]
and perform each indicated row operation:
a. R_1 \leftrightarrow R_2 b. \frac{1}{3} R_1 c. 2 R_2+R_3.
a. The notation R_1 \leftrightarrow R_2 means to interchange the elements in row 1 and row 2 . This results in the row-equivalent matrix
b. The notation \frac{1}{3} R_1 means to multiply each element in row 1 by \frac{1}{3}. This results in the row-equivalent matrix
\left[\begin{array}{ccc|c}\frac{1}{3}(3) & \frac{1}{3}(18) & \frac{1}{3}(-12) & \frac{1}{3}(21) \\1 & 2 & -3 & 5 \\-2 & -3 & 4 & -6\end{array}\right]=\left[\begin{array}{rrr|r}1 & 6 & -4 & 7 \\1 & 2 & -3 & 5 \\-2 & -3 & 4 & -6\end{array}\right]
c. The notation 2 R_2+R_3 means to add 2 times the elements in row 2 to the corresponding elements in row 3 . Replace the elements in row 3 by these sums. First, we find 2 times the elements in row 2 , namely, 1, 2, -3, and 5 :
2(1) or 2, 2(2) or 4, 2(–3) or –6, 2(5) or 10.
Now we add these products to the corresponding elements in row 3 . Although we use row 2 to find the products, row 2 does not change. It is the elements in row 3 that change, resulting in the row-equivalent matrix
