Determine the rank of the matrix $D=\left(\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right)$ and state whether the system $D\mathbf{x}=\left(\begin{array}{c}5\\ 10\\ 15\end{array}\right)$ is consistent.

Step-by-step solution

1. Observe that each row of $D$ is a multiple of the first row: Row2 = 2·Row1, Row3 = 3·Row1, so all rows are linearly dependent.
2. Therefore the maximum number of independent rows is 1, giving $\mathrm{rank}(D)=1$.
3. Form the augmented matrix $[D|\mathbf{b}]=\left(\begin{array}{ccccc}1& 2& 3& |& 5\\ 2& 4& 6& |& 10\\ 3& 6& 9& |& 15\end{array}\right)$.
4. Since the RHS vector $\mathbf{b}$ is also a multiple of the first row (5·Row1), the augmented matrix has the same rank as $D$, i.e., 1.
5. Because $\mathrm{rank}(D)=\mathrm{rank}([D|\mathbf{b}])<\text{number of unknowns (3)}$, the system is consistent and has infinitely many solutions.
6. Thus the rank is 1 and the system is consistent with infinitely many solutions.

Concept tested: Rank of a matrix and consistency of a linear system

Published 6 September 2026 · How this solution was prepared and checked