Question

Please show all steps in completing this problem, thank you very much!

Solve the system Ax=b using the LU factorization of A and the matrix b given below. -2 0 0 1 -1 -2 7 A=LU= -2 -1 0 0 1 3 -17 2 -1 -2 0 0 1 -5 b= 12 12 -10 You can resize a matrix (when appropriate) by clicking and dragging the bottom-right corner of the matrix. The system has infinitely many solutions Number of Parameters: 1 [x1] [o] [o *4 0 0

Answer 1

here LU factorization is given so first we solve LY=B

-2 0 0 12 y1 -2 -1 0 12 y2 -1-2 2 -10

24112

y1=-6

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2/1 y2 12

2(-6) y2= 12

12 y2 12

y2=12- 12

y2=0

2=0

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2y1-y2-23 = -10

2(-6) 0 2y3-10

$-12-2y_3=-10$

$-2y_3=-10+12$

2 243

${\color{Blue} y_3=-1}$

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vector y is

${\color{Blue} \begin{pmatrix}y_1\\ y_2\\ y_3\end{pmatrix}=\begin{pmatrix}-6\\ 0\\ -1\end{pmatrix}}$

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now solve for UX=Y

$\begin{pmatrix}1&-1&-2&7\\ 0&1&3&-17\\ 0&0&1&-5\end{pmatrix}\begin{pmatrix}x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix}-6\\ 0\\ -1\end{pmatrix}$

$x_3-5x_4=-1$

${\color{Blue} x_3=-1+5x_4}$

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3r3-17x4

$x_2+3(-1+5x_4)-17x_4=0$

$x_2-3+15x_4-17x_4=0$

$x_2-2x_4-3=0$

${\color{Blue} x_2 =3+2x_4}$

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$x_1-x_2-2x_3+7x_4=-6$

$x_1-(3+2x_4)-2(-1+5x_4)+7x_4=-6$

$x_1-3-2x_4+2-10x_4+7x_4=-6$

1-5x4- 1 = -6

$x_1=-6+1+5x_4$

${\color{Blue} x_1=-5+5x_4}$

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${\color{Blue} x_4=x_4}.................free$

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general solution is

$\begin{pmatrix}x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix}-5\\ 3\\ -1\\ 0\end{pmatrix}+x_4\begin{pmatrix}5\\ 2\\ 5\\ 1\end{pmatrix}$

take $x_4=s$

${\color{Red} \begin{pmatrix}x_1\\ x_2\\ x_3\\ x_4\end{pmatrix}=\begin{pmatrix}-5\\ 3\\ -1\\ 0\end{pmatrix}+s\begin{pmatrix}5\\ 2\\ 5\\ 1\end{pmatrix}}$