Answer to Question #198921 in Linear Algebra for FELICIA NDANGUMUNI

Question #198921

Consider the following augmented matrix 1 -1 2 1

3 -1 5 -2

-4 2 2x^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


1
Expert's answer
2021-05-27T06:03:46-0400

Let A = [11213152422x28x+2]\begin{bmatrix} 1 & -1 & 2 & 1\\ 3 & -1 & 5 & -2\\ -4 & 2 & 2x^2-8 & x+2 \end{bmatrix}





R2=R23R1R_2=R_2-3R_1



[1121021542x28x+2]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ -4 & 2 & x^2-8 & x+2 \end{bmatrix}

R3=R3+4R1R_3=R_3+4R_1



[1121021502x2x+6]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 2 & -1 & -5 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}

R2=R2/2R_2=R_2/2



[1121011/25/202x2x+6]\begin{bmatrix} 1 & -1 & 2 & 1 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}

R1=R1+R2R_1=R_1+R_2




[103/23/2011/25/202x2x+6]\begin{bmatrix} 1 & 0 & 3/2 & -3/2 \\ 0 & 1 & -1/2 & -5/2 \\ 0 & -2 & x^2 & x+6 \end{bmatrix}

R3=R3+2R2R_3=R_3+2R_2




[103/23/2011/25/200x21x+1]\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



x21=0x+10=>x=1\begin{matrix} x^2-1=0 \\ x+1\not=0 \end{matrix}=>x=1

Ii) exactly one solution



x210=>x±1x^2-1\not=0=>x\not=\pm1

Iii) infinitely many solutions



x21=0x+1=0=>x=1\begin{matrix} x^2-1=0 \\ x+1=0 \end{matrix}=>x=-1

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