Chapter 8, Problem 1RE

To determine

To calculate: Perform the operation $\text{-5}{R}_2+R_3$ on $\left[\begin{array}{cccc}2& -2& 1& -1\\ 1& 2& -1& 2\\ 6& 4& 3& 5\end{array}\right]$ and write new matrix.

Answer to Problem 1RE

Solution: The new matrix,

$\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0& 0& 9& -9\end{array}\right]$

Explanation of Solution

Given Information: The matrix is $\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0& 5& 4& 1\end{array}\right]$ and the operation is, $\text{-5}{R}_2+R_3$ in the second row.

Calculation:

Consider the given matrix,

$\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0& 5& 4& 1\end{array}\right]$

Perform the operation, $-5R_2+R_3$

For this we will first multiply row 2 by $\left(-5\right)$ to get,

$\begin{array}{c}0\left(-5\right)=0\\ 1\left(-5\right)=-5\\ -1\left(-5\right)=5\\ 2\left(-5\right)=-10\end{array}$

Now adding this with row 3 as below, we get:

$\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0+0& 5+\left(-5\right)& 4+5& 1+\left(-10\right)\end{array}\right]=\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0& 0& 9& -9\end{array}\right]$

The new matrix is: $\left[\begin{array}{cccc}1& 2& 2& 2\\ 0& 1& -1& 2\\ 0& 0& 9& -9\end{array}\right]$