Chapter 7.4, Problem 5E

To determine

Whether the augmented matrixof the given system of linear equation has dimension $3\times 2$ .

Answer to Problem 5E

The augmented matrix of the given system of linear equation has dimension $2\times 3$ .

Explanation of Solution

Given information:

A system of linear equation is given as

  $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$

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.

  $\{\begin{array}{l}-2x+3y=5\\ 6x+7y=4\end{array}$

Now, the augmented matrix is given by

  $\left[\begin{array}{cccc}-2& 3& ⋮& 5\\ 6& 7& ⋮& 4\end{array}\right]$

Here, the number of row of the matrix is 2 and number of columns is 3.

So, dimension of the matrix is

  $\begin{array}{c}\text{Dimension}=\text{Number}\text{of}\text{rows}\times \text{Number}\text{of}\text{columns}\\ =2\times 3\end{array}$

Hence, the augmented matrix of the system of linear equation don’t have a dimension of $3\times 2$ .