For the matrix B=\begin{pmatrix}1&2&3\2&4&6\3&5&9\end{\pmatrix}, determine the rank and decide whether the system Bmathbf{x}=\begin{pmatrix}7\14\21\end{\pmatrix} is consistent. If consistent, give the general solution.

Step-by-step solution

1. Compute $\det B$ using cofactor expansion: \det B=1\begin{vmatrix}4&6\5&9\end{\vmatrix}-2\begin{vmatrix}2&6\3&9\end{\vmatrix}+3\begin{vmatrix}2&4\3&5\end{\vmatrix}=0. Since $\det B=0$, $B$ is singular and its rank is at most 2.
2. Check linear independence of rows. Row 2 is $2\times$ Row 1, but Row 3 is not a multiple of Row 1. Hence rank$=2$.
3. Form the augmented matrix and perform row reduction: \begin{pmatrix}1&2&3&|&7\2&4&6&|&14\3&5&9&|&21\end{\pmatrix}\nSubtract $2\times$ Row 1 from Row 2 → Row 2 becomes all zeros. Subtract $3\times$ Row 1 from Row 3 → Row 3 becomes $(0,-1,0|0)$.
4. The rank of the coefficient matrix is 2 and the rank of the augmented matrix is also 2, so the system is consistent with infinitely many solutions.
5. Solve for free variable $z=t$: From Row 3: $-y=0\Rightarrow y=0$. From Row 1: $x+2y+3z=7\Rightarrow x+3t=7\Rightarrow x=7-3t$.
6. Thus the solution set is $(x,y,z)=(7-3t,0,t)$ where $t$ is any real number.

Concept tested: Rank, consistency, and general solution of a linear system

Published 13 October 2026 · How this solution was prepared and checked