For the matrix $B = 123246369$ and the vector $b = 61218$ , determine the rank of $B$ and state whether the linear system $B x = b$ is consistent. If consistent, give the general solution.

medium 3 marks Class 12 · Mathematics · Matrices

Answer

Infinitely many solutions;

x = t 123, t \in R

The correct answer is Infinitely many solutions; $x = t 123, t \in R$ — Solve by parameterizing: Let $x = t$ . From first row: $t + 2 y + 3 z = 6$ . Express $y$ and $z$ as multiples of $t$ . The simplest representation is $x = t 123$ .

Step-by-step solution

Check if rows are multiples: row2=2×row1, row3=3×row1. So rank of $B$ is 1.
Form augmented matrix $[B ∣ b]$ .\nSince $b$ is also a multiple of row1 ( $b = 6 \times 123$ ), the augmented matrix has the same rank 1.
Ranks of coefficient and augmented matrices are equal; therefore system is consistent.
Because $rank (B) = 1 < 3$ , there are infinitely many solutions.
Solve by parameterizing: Let $x = t$ . From first row: $t + 2 y + 3 z = 6$ . Express $y$ and $z$ as multiples of $t$ . The simplest representation is $x = t 123$ .

Common mistake: Assuming only zero solution because coefficient matrix has rank 1

Concept tested: Rank and consistency of a linear system

Published 17 September 2026 · How this solution was prepared and checked

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For the matrix B=​123​246​369​​ and the vector b=​61218​​, determine the rank of B and state whether the linear system Bx=b is consistent. If consistent, give the general solution.

Step-by-step solution

For the matrix $B = 123246369$ and the vector $b = 61218$ , determine the rank of $B$ and state whether the linear system $B x = b$ is consistent. If consistent, give the general solution.