UPSC 2024 Maths Optional Paper 2 Q5b — Step-by-Step Solution

10 marks · Section B

Gauss-Jordan method · Numerical Analysis · asked 2× in 13 yrs · Read the full method →

Question

Solve the following system by the Gauss-Jordan method:

2x+3y-z=5,\quad 4x+4y-3z=3,\quad 2x-3y+2z=2.

Forward elimination to upper-triangular form, then back-elimination to identity (Gauss-Jordan = continue past upper-triangular).

Augmented matrix:

[A|b]=\begin{bmatrix}2 & 3 & -1 & | & 5\\ 4 & 4 & -3 & | & 3\\ 2 & -3 & 2 & | & 2\end{bmatrix}.

Forward elimination:

$R_2\to R_2-2R_1$ : $(0,-2,-1,-7)$ . $R_3\to R_3-R_1$ : $(0,-6,3,-3)$ . $R_3\to R_3-3R_2$ : $(0,0,6,18)$ .

Normalise pivots:

$R_3\to R_3/6$ : $(0,0,1,3)$ . $R_2\to R_2/(-2)$ : $(0,1,1/2,7/2)$ . $R_1\to R_1/2$ : $(1,3/2,-1/2,5/2)$ .

Back-elimination:

$R_2\to R_2-(1/2)R_3$ : $(0,1,0,2)$ . $R_1\to R_1+(1/2)R_3$ : $(1,3/2,0,4)$ .

$R_1\to R_1-(3/2)R_2$ : $(1,0,0,1)$ .

Reduced augmented matrix:

\begin{bmatrix}1 & 0 & 0 & | & 1\\ 0 & 1 & 0 & | & 2\\ 0 & 0 & 1 & | & 3\end{bmatrix}.

\boxed{\;x=1,\;y=2,\;z=3.\;}