Chapter 7.4, Problem 105E

To determine

The nature of a given system of linear equations derived from a given augmented matrix.

Answer to Problem 105E

A $2\times 4$ augmented matrix represents a dependent system of linear equations since one variable can only be represented as a combination of the other variables.

Explanation of Solution

Given: A $2\times 4$ augmented matrix with all non-zero real numbers.

Concept Used: An augmented matrix is a matrix representation of a standard system of simultaneous linear equations. It is a matrix which consists of two parts with a line separating both parts. On the left hand side of the line is the coefficient matrix (generally denoted by A ) of the system of equations and on the right hand side of the line are the constant terms (generally denoted by B ), written in order. Multiplying the coefficient matrix part to the variable column matrix (generally denoted by X) gives the algebraic expressions which are then equated to the constant terms to get the system of equations. Thus, the system of equations can be found using the matrix equation-

  $AX=B$

For solving any system of linear equations, the number of variables much be equal to the number of equations, otherwise the system is inconsistent and the variables will have infinitely many values.

Consider a general $2\times 4$ augmented matrix, say $\left[\begin{array}{cccc}{a}_1& b_1& c_1& d_1\\ a_2& b_2& c_2& d_2\end{array}|\begin{array}{c}p\\ q\end{array}\right]$ .

The coefficient matrix and the constant matrix associated with this augmented matrix are-

  $A=\left[\begin{array}{cccc}{a}_1& b_1& c_1& d_1\\ a_2& b_2& c_2& d_2\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]$ respectively.

The variable matrix will consist of 4 variables viz. w , x , y , and z . Thus the variable matrix will be-

  $X=\left[\begin{array}{c}w\\ x\\ y\\ z\end{array}\right]$

Hence, the required system of equations associated with this augmented matrix is given by-

  $AX=B$

  $\Rightarrow \left[\begin{array}{cccc}{a}_1& b_1& c_1& d_1\\ a_2& b_2& c_2& d_2\end{array}\right]\left[\begin{array}{c}w\\ x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}p\\ q\end{array}\right]$

  $\Rightarrow \left[\begin{array}{c}{a}_{1}w+b_{1}x+c_{1}y+d_{1}z\\ a_{2}w+b_{2}x+c_{2}y+d_{2}z\end{array}\right]=\left[\begin{array}{c}p\\ q\end{array}\right]$

  $\Rightarrow a_{1}w+b_{1}x+c_{1}y+d_{1}z=p$ . . . . . . . ( i )

  $a_{2}w+b_{2}x+c_{2}y+d_{2}z=q$ . . . . . . . ( ii )

Since there are 4 variables but only two linear equations, thus this system is inconsistent. Hence, the values of one variable can be derived as a combination of the two other variables which implies the system of equations is dependent. In general, if the coefficient matrix part of any augmented matrix is not a square matrix, then the system of linear equations is dependent.