← 2016 Paper 1

UPSC 2016 Maths Optional Paper 1 Q1b-i — Step-by-Step Solution

7 marks · Section A

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

Question

Using elementary row operations, find the condition that the linear equations

x−2y+z=a2x+7y−3z=b3x+5y−2z=c\begin{aligned}x-2y+z&=a\\ 2x+7y-3z&=b\\ 3x+5y-2z&=c\end{aligned}

have a solution.

Technique

Row-reduce [A ∣ b][A\,|\,\mathbf b]; when the coefficient part develops a zero row, the matching right-hand entry must vanish for consistency (Rouché–Capelli).

Solution

Step 1 — Form the augmented matrix

(1−21a27−3b35−2c).\left(\begin{array}{ccc|c}1&-2&1&a\\2&7&-3&b\\3&5&-2&c\end{array}\right).

Step 2 — Eliminate the first column

R2→R2−2R1R_2\to R_2-2R_1 and R3→R3−3R1R_3\to R_3-3R_1:

(1−21a011−5b−2a011−5c−3a).\left(\begin{array}{ccc|c}1&-2&1&a\\0&11&-5&b-2a\\0&11&-5&c-3a\end{array}\right).

Step 3 — Eliminate the second column from R3R_3

R3→R3−R2R_3\to R_3-R_2:

(1−21a011−5b−2a000 c−3a−(b−2a) )=(1−21a011−5b−2a000 c−a−b ).\left(\begin{array}{ccc|c}1&-2&1&a\\0&11&-5&b-2a\\0&0&0&\,c-3a-(b-2a)\,\end{array}\right)=\left(\begin{array}{ccc|c}1&-2&1&a\\0&11&-5&b-2a\\0&0&0&\,c-a-b\,\end{array}\right).

Step 4 — Read off the consistency condition

The coefficient matrix has rank 22 (third row of coefficients is all zero). The system is consistent iff the augmented matrix also has rank 22, i.e. the last right-hand entry vanishes:

c−a−b=0.c-a-b=0.

Answer

  The system has a solution  ⟺  a+b=c.  \boxed{\;\text{The system has a solution}\iff a+b=c.\;}
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