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UPSC 2017 Maths Optional Paper 1 Q4b — 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

Consider the following system of equations in $x,y,z$:

• (i) For which values of $a$ does the system have a unique solution?
• (ii) For which pair of values $(a,b)$ does the system have more than one solution?

Technique

$\Delta\ne0\Rightarrow$ unique; for $\Delta=0$ row-reduce the augmented matrix and impose consistency to find the $(a,b)$ giving infinitely many solutions.

Solution

Step 1 — Determinant of the coefficient matrix

Subtract $R_1$ from $R_2$ and $R_3$:

Step 2 — (i) Unique solution

A $3\times3$ system has a unique solution iff $\Delta\ne0$:

Step 3 — (ii) More than one solution requires $\Delta=0$ and consistency

More-than-one (in fact infinitely many) solutions require $\Delta=0$, i.e. $a=5$ or $a=-1$, and the augmented system to be consistent. Eliminate using $R_2'=R_2-R_1$, $R_3'=R_3-R_1$:

Eliminate to expose the consistency condition: $(a-2)\,R_2'-9\,R_3$ combination gives the last-row residual.

• Case $a=-1$: $R_2':\ -3y+z=2$, $R_3':\ 9y-3z=b-1$. Note $R_3'=-3R_2'$ on the left, so the left sides are proportional; consistency needs $b-1=-3(2)=-6$, i.e. $b=-5$. The carried-out elimination gives the residual $-3b-15=0\Rightarrow b=-5$.
• Case $a=5$: $R_2':\ 3y+z=2$, $R_3':\ 9y+3z=b-1$, and $R_3'=3R_2'$ on the left, so consistency needs $b-1=3(2)=6$, i.e. $b=7$. The residual is $3b-21=0\Rightarrow b=7$.

Hence

Answer