UPSC 2022 Maths Optional Paper 1 Q2a — Step-by-Step Solution

15 marks · Section A

Solution of system of linear equations · Linear Algebra · asked 7× in 13 yrs · Read the full method →

Question

Find all solutions to the system by row-reduced method:

x_1+2x_2-x_3=2

2x_1+3x_2+5x_3=5

-x_1-3x_2+8x_3=-1

Gaussian elimination to row-reduced echelon form; identify free parameter (since rank < #unknowns); parametrise.

\left(\begin{array}{ccc|c}1 & 2 & -1 & 2\\ 2 & 3 & 5 & 5\\ -1 & -3 & 8 & -1\end{array}\right).

$R_2\to R_2-2R_1$ , $R_3\to R_3+R_1$ :

\left(\begin{array}{ccc|c}1 & 2 & -1 & 2\\ 0 & -1 & 7 & 1\\ 0 & -1 & 7 & 1\end{array}\right).

$R_3\to R_3-R_2$ :

\left(\begin{array}{ccc|c}1 & 2 & -1 & 2\\ 0 & -1 & 7 & 1\\ 0 & 0 & 0 & 0\end{array}\right).

$R_2\to -R_2$ :

\left(\begin{array}{ccc|c}1 & 2 & -1 & 2\\ 0 & 1 & -7 & -1\\ 0 & 0 & 0 & 0\end{array}\right).

$R_1\to R_1-2R_2$ :

\left(\begin{array}{ccc|c}1 & 0 & 13 & 4\\ 0 & 1 & -7 & -1\\ 0 & 0 & 0 & 0\end{array}\right).

Rank = 2, three unknowns, so one free parameter. Let $x_3=t$ .

From row 1: $x_1+13t=4\Rightarrow x_1=4-13t$ . From row 2: $x_2-7t=-1\Rightarrow x_2=-1+7t$ .

\boxed{\;(x_1,x_2,x_3)=(4-13t,\;-1+7t,\;t),\quad t\in\mathbb R.\;}