For the system $x + 2 y - z = 3$ , $2 x + 4 y - 2 z = 6$ , $3 x + 6 y - 5 z = 2$ , find the rank of the coefficient matrix and the augmented matrix and state whether the system is consistent. If it is consistent, give the solution.

medium 3 marks Class 12 · Mathematics · Matrices

Answer

No solution (the system is inconsistent)

The correct answer is No solution (the system is inconsistent) — Since $rank A \neq = rank [A ∣ B]$ , the system is inconsistent and has no solution.

Step-by-step solution

Write the coefficient matrix $A = 123246 - 1 - 2 - 5$ and the augmented matrix $[A ∣ B] = 123246 - 1 - 2 - 5 ∣ ∣ ∣ 362$ .
Perform $R_{2} \leftarrow R_{2} - 2 R_{1}$ giving $R_{2} = (0, 0, 0∣0)$ .
Perform $R_{3} \leftarrow R_{3} - 3 R_{1}$ giving $R_{3} = (0, 0, - 2∣ - 7)$ .
Now $R_{3}$ has a non‑zero entry in the coefficient part, so the rank of $A$ is $2$ .
The augmented part of $R_{3}$ is $- 7$ , not zero, therefore the rank of $[A ∣ B]$ is $3$ .
Since $rank A \neq = rank [A ∣ B]$ , the system is inconsistent and has no solution.

Common mistake: Assuming a zero row in the coefficient matrix automatically gives a solution without checking the corresponding RHS entry.

Concept tested: Rank of matrix and consistency of linear system

Published 29 August 2026 · How this solution was prepared and checked

For the system x+2y−z=3, 2x+4y−2z=6, 3x+6y−5z=2, find the rank of the coefficient matrix and the augmented matrix and state whether the system is consistent. If it is consistent, give the solution.

Step-by-step solution

Related questions

For the system $x + 2 y - z = 3$ , $2 x + 4 y - 2 z = 6$ , $3 x + 6 y - 5 z = 2$ , find the rank of the coefficient matrix and the augmented matrix and state whether the system is consistent. If it is consistent, give the solution.