Chapter 8.2, Problem 60E

(A)

To determine

To find:

The given system of linear equations in the form of matrix equation

Answer to Problem 60E

The form of matrix equation is $\left[\begin{array}{cc}-4& 9\\ 1& -3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}-13\\ 12\end{array}\right]$

Explanation of Solution

Given:

The system of linear equation

  $\begin{array}{l}-4x_1+9x_2=-13\\ x_1-3x_2=12\end{array}$

Concept used:

The Matrix equation

  $AX=B$

Where A is the coefficient matrix of the system

  $X$ And $B$ are column matrices

The column matrix $B$ is also called a constant matrix its entries are the constant terms in the system of equation

Calculation:

The system of linear equation

  $\begin{array}{l}-4x_1+9x_2=-13\\ x_1-3x_2=12\end{array}$

The Matrix equation

  $AX=B\mathrm{.................}\left(1\right)$

Where A is the coefficient matrix of the system

  $X$ And $B$ are column matrices

  $A=\left[\begin{array}{cc}-4& 9\\ 1& -3\end{array}\right],X=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],B=\left[\begin{array}{c}-13\\ 12\end{array}\right]$

From equation (1)

  $\left[\begin{array}{cc}-4& 9\\ 1& -3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}-13\\ 12\end{array}\right]$

(B)

To determine

To find:

The matrix by using the Gauss-Jordan elimination on augmented matrix $\left[A:B\right]$

Answer to Problem 60E

The Gauss-Jordan elimination on augmented matrix $A=\left[\begin{array}{c}12\\ 35\end{array}\right]$

Explanation of Solution

Given:

The system of linear equation

  $\begin{array}{l}-4x_1+9x_2=-13\\ x_1-3x_2=12\end{array}$

Concept used:

The Gauss-Jordan elimination on augmented matrix $\left[A:B\right]$

:-An elementary row operations are applied to a matrix to obtain a (row-equivalent) row echelon form of the matrix. A second method of elimination is called Gauss-Jordan elimination

Calculation:

The system of linear equation

  $\begin{array}{l}-4x_1+9x_2=-13\\ x_1-3x_2=12\end{array}$

The Matrix equation

  $AX=B\mathrm{.................}\left(1\right)$

Where A is the coefficient matrix of the system

  $X$ And $B$ are column matrices

  $A=\left[\begin{array}{cc}-4& 9\\ 1& -3\end{array}\right],X=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right],B=\left[\begin{array}{c}-13\\ 12\end{array}\right]$

The augmented matrix

  $\begin{array}{l}A=\left[\begin{array}{l}\begin{array}{c}\begin{array}{c}-4\end{array}\end{array}\begin{array}{c}9\end{array}:-13\\ \begin{array}{c}\begin{array}{c}1\end{array}\end{array}\begin{array}{c}-3\end{array}:\begin{array}{c}12\end{array}\end{array}\right]\\ R_1\leftrightarrow R_2\\ A=\left[\begin{array}{l}\begin{array}{c}\begin{array}{c}1\end{array}\end{array}\begin{array}{c}-3\end{array}:12\\ \begin{array}{c}\begin{array}{c}-4\end{array}\end{array}\begin{array}{c}9\end{array}:\begin{array}{c}-13\end{array}\end{array}\right]\\ R_2\to R_2+4R_1\\ A=\left[\begin{array}{l}\begin{array}{c}\begin{array}{c}1\end{array}\end{array}\begin{array}{c}-3\end{array}:12\\ \begin{array}{c}\begin{array}{c}0\end{array}\end{array}\begin{array}{c}-3\end{array}:\begin{array}{c}35\end{array}\end{array}\right]\\ A=\left[\begin{array}{c}12\\ 35\end{array}\right]\end{array}$