Chapter 7.4, Problem 16E

To determine

To write: the augmented matrixof the given system of linear equation and its dimension.

Answer to Problem 16E

Augmented matrix of the system of equation is

  $\left[\begin{array}{ccccc}1& -3& 1& ⋮& 1\\ 0& 4& 0& ⋮& 0\\ 0& 0& -7& ⋮& -5\end{array}\right]$

The augmented matrix of the given system of linear equation has dimension $3\times 4$ .

Explanation of Solution

Given information:

A system of linear equation is given as

  $\{\begin{array}{l}x-3y+z=1\\ 4y=0\\ 7z=-5\end{array}$

Concept used:

Co-efficient matrix of a system of linear equation of the form $\{\begin{array}{l}{a}_{1}x+b_{1}y=c_1\\ a_{2}x+b_{2}y=c_2\end{array}$ is given by

  $\left[\begin{array}{cc}{a}_1& b_1\\ a_2& b_2\end{array}\right]$

The augmented matrix of the system of linear equation is given by

  $\left[\begin{array}{cccc}{a}_1& b_1& ⋮& c_1\\ a_2& b_2& ⋮& c_2\end{array}\right]$

Dimension of a matrix is

  $\text{Dimension}=\text{Number}\text{of}\text{rows}\times \text{Number}\text{of}\text{columns}$

Consider the given system of equation.

  $\{\begin{array}{l}x-3y+z=1\\ 4y=0\\ 7z=-5\end{array}$

Now, the augmented matrix is given by

  $\left[\begin{array}{ccccc}1& -3& 1& ⋮& 1\\ 0& 4& 0& ⋮& 0\\ 0& 0& -7& ⋮& -5\end{array}\right]$

Here, the number of row of the matrix is 3 and number of columns is 4.

So, dimension of the matrix is

  $\begin{array}{c}\text{Dimension}=\text{Number}\text{of}\text{rows}\times \text{Number}\text{of}\text{columns}\\ =3\times 4\end{array}$

The augmented matrix of the given system of linear equation has dimension $3\times 4$ .