Question

Gauss-Seidel method is used to solve the following equations (as per the given order):
$x_1+2x_2+3x_3=5$,
$2x_1+3x_2+x_3=1$
$3x_1+2x_2+x_3=3$
Assuming initial guess as $x_1=x_2=x_3=0$, the value of $x_3$ after the first iteration is

• A. -6

Answer 1

The correct option is A -6
Given that
$x_1=x_2=x_3=0$
Hence,
$x_1+2\left(0\right)+3\left(0\right)=5$
$x_1=5$
Now for $x_2$, we have $x_1=5$ & $x_3=0$
i.e.,2$x_1+3x_2+0=1$
$\Rightarrow$ 3$x_2=1-2x_1$
$\Rightarrow$ 3$x_2=1-2\left(5\right)=-9$
$\because$ $x_2=-3$
Now for $x_3$, we will take $x_1=5$ & $x_2=-3$
i.e.,3$x_1+2x_2+x_3=3$
$x_3=3-3x_1-2x_2$
$x_3=3-3\left(5\right)-2(-3)=-6$
Hence, -6 is correct.
Note: Here we have not changed given system into diagonally dominant system as it is mentioned that we have to solve this system "As per the given order".