Q. 4.78 - Intermediate Algebra

Question

Perform the indicated operations on the augmented matrix: $\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

$\left(a\right)$Interchange rows $1$ and $2$.

$\left(b\right)$Multiply row $1$ by $2$.

$\left(c\right)$Multiply row $2$ by $3$ and add to row $1$.

Step-by-Step Solution

Answer

Part $\left(a\right)$ Interchange row $1$ and $2$ :-

$\left[\begin{array}{rrrr}4& 1& -3& 2\\ 2& -3& -2& -4\\ 5& 0& 4& -1\end{array}\right]$

Part $\left(b\right)$ Multiply row $1$ by $2$ :-

$\left[\begin{array}{rrrr}4& -6& -4& -8\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

Part $\left(c\right)$ Multiply row $2$ by $3$ and add to row $1$

$\left[\begin{array}{rrrr}14& 0& -11& 2\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

Part (a) Step 1. Interchange the rows 1 and 2 .

Given augmented matrix is $\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

We interchange row $1$ and $2$

$\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right] \stackrel{R_1\leftrightarrow R_2}{\to } \left[\begin{array}{rrrr}4& 1& -3& 2\\ 2& -3& -2& -4\\ 5& 0& 4& -1\end{array}\right]$

Part (b) Step 1. Multiply row 1 by 2 .

Given augmented matrix is $\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

We multiply row $1$ by $2$

$\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right] \stackrel{2R_1}{\to } \left[\begin{array}{rrrr}4& -6& -4& -8\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

Part (c) Step 1. Multiply row 2 by 3 and add to row 1 .

Given augmented matrix is $\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

We multiply row $2$ by $3$ and add to row $1$

$\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right] \stackrel{3R_2+R_1}{\to } \left[\begin{array}{rrrr}(3*4)+2& (3*1)-3& (3*(-3\left)\right)-2& (3*2)-4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$

$\left[\begin{array}{rrrr}2& -3& -2& -4\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right] \stackrel{3R_2+R_1}{\to } \left[\begin{array}{rrrr}14& 0& -11& 2\\ 4& 1& -3& 2\\ 5& 0& 4& -1\end{array}\right]$