First slide

System of linear equations

Question

The system of equations $x + 4 y - 2 z = 3$ , $3 x + y + 5 z = 7$ and $2 x + 3 y + z = 5$ has

Moderate

A

Infinite no. of solutions
B

Unique solution
C

Trivial solution
D

No solution

Solution

$\begin{array}{l} Augmented Matrix = [\begin{matrix} 1 & 4 & - 2 & 3 \\ 3 & 1 & 5 & 7 \\ 2 & 3 & 1 & 5 \end{matrix}] \\ Apply R_{2} \to R_{2} - 3 R_{1}; R_{3} \to R_{3} - 2 R_{1} \\ ~ [\begin{matrix} 1 & 4 & - 2 & 3 \\ 0 & - 11 & 11 & - 2 \\ 0 & - 5 & 5 & - 1 \end{matrix}] \\ Apply R_{3} \to 11 R_{3} - 5 R_{2} \\ ~ [\begin{matrix} 1 & 4 & - 2 & 3 \\ 0 & - 11 & 11 & - 2 \\ 0 & 0 & 0 & - 1 \end{matrix}] \end{array}$
Thus, rank(A)=2 ; rank(AB)=3
Thus, rank(A) $\neq$ rank(AB)
Thus, the system is inconsistent, no solution.

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