Answer to Question #196382 in Linear Algebra for Rebecca

Question

Consider the following augmented matrix [ 1 -1 2 1]

[3 -1 5 -2]

[-4 2 x^2-8 x+2]

Determine the values of x for which the system has

I) no solution,

Ii) exactly one solution,

Iii) infinitely many solutions

Answer 1

Augmented matrix 

$$A=\begin{bmatrix} 1 & -1 & 2 & 1 \\ 3 & -1 & 5 & -2 \\ -4 & 2 & x^2-8 & x+2 \end{bmatrix}$$

$R_2=R_2-3R_1$

$$\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ -4 & 2 & x^2-8 & x+2 \end{bmatrix}$$

$R_3=R_3+4R_1$

$$\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}$$

$R_2=R_2/2$

$$\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}$$

$R_1=R_1+R_2$

$$\begin{bmatrix} 1 & 0 & 3/2 & -3/2 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}$$

$R_3=R_3+2R_2$

$$\begin{bmatrix} 1 & 0 & 3/2 & -3/2 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & 0 & x^2-1 & x+1 \end{bmatrix}$$

I) no solution

$$\begin{matrix} x^2-1=0 \\ x+1\not=0 \end{matrix}=>x=1$$

Ii) exactly one solution

$$x^2-1\not=0=>x\not=\pm1$$

Iii) infinitely many solutions

$$\begin{matrix} x^2-1=0 \\ x+1=0 \end{matrix}=>x=-1$$