Question

What is the difficulty in applying Gauss elimination to the

following linear algebraic equations?

$$\begin{gathered} \mathbf{-}\mathbf{5}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{5}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{5} \\ \mathbf{10}{\mathbf{x}}_{\mathbf{1}}\mathbf{-}\mathbf{10}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{5} \end{gathered}$$

Short answer

Therefore, the solution is not possible.

Step-by-step solution

Given data.

The given linear equations are,

$$\begin{gathered} -5x_1+5x_2=5.....\left(1\right) \\ 10x_1-10x_2=-5.....\left(2\right) \end{gathered}$$

Determine the formulas of linear equation.

Write the formula of general form of linear equation.

$\mathbf{Ax}\mathbf{=}\mathbf{B}$ ……. (3)

Here, x is the$N\times 1$matrix which is the required solution, A is a$N\times N$matrix

which represents the coefficients of x , and$B$is the$N\times 1$matrix.

Here, Nis order of the system.

Apply the Gauss Elimination method.

Write the equations (1) and (2) in matrix form.

$\left[\begin{array}{cc}-5& 5\\ 10& -10\end{array}\right]\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right]=\left[\begin{array}{c}5\\ -5\end{array}\right]$ ……. (4)

For Gauss Siedel elimination method,$N-1$steps are needed.

From matrix (4), the order of the matrix is 2 , therefore,

$$\begin{gathered} N-1=2-1 \\ =1 \end{gathered}$$

Step 1: Now, in equation (4), multiply first row by 2 and add them to rows 2 .

$\left[\begin{array}{cc}-5& 5\\ 0& 0\end{array}\right]\left[\begin{array}{c}{\mathrm{x}}_1\\ {\mathrm{x}}_2\end{array}\right]=\left[\begin{array}{c}5\\ 5\end{array}\right]$

Solve linear equations.

The second row says that,

$-0{\mathrm{x}}_1+0{\mathrm{x}}_2=5$ ……. (5)

The equation (5) is not possible, because determinant of$\left|\begin{array}{cc}-5& 5\\ 10& -10\end{array}\right|$is zero.

Therefore, the solution is not possible.