For the matrix $B=\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 4& 6& 8\\ 0& 1& 1& 0\end{array}\right)$ and the column vector $\mathbf{b}=\left(\begin{array}{c}10\\ 20\\ 3\end{array}\right)$, determine (i) the rank of the coefficient matrix, (ii) the rank of the augmented matrix $[B|\mathbf{b}]$, and (iii) whether the linear system $B\mathbf{x}=\mathbf{b}$ is consistent. If consistent, state the nature of the solution set.

Step-by-step solution

1. Observe that the second row of $B$ is twice the first row: $R_2=2R_1$, so it does not increase rank.
2. The first row $R_1=(1,2,3,4)$ and the third row $R_3=(0,1,1,0)$ are not scalar multiples; they are linearly independent.
3. Hence the rank of $B$ is $2$.
4. Form the augmented matrix $[B|\mathbf{b}]=\left(\begin{array}{cccccc}1& 2& 3& 4& |& 10\\ 2& 4& 6& 8& |& 20\\ 0& 1& 1& 0& |& 3\end{array}\right)$.
5. Row $2$ of the augmented matrix is also twice row $1$ (including the constant term), so it adds no new information.
6. Rows $1$ and $3$ remain independent; therefore rank $[B|\mathbf{b}]=2$.
7. Since the ranks of the coefficient and augmented matrices are equal (both $2$) and are less than the number of variables ($4$), the system is consistent and has infinitely many solutions.

Concept tested: Rank of matrix and consistency of linear system

Published 15 September 2026 · How this solution was prepared and checked