Find the rank of the matrix $P=\left(\begin{array}{ccc}1& 2& 4\\ 2& 4& 8\\ 3& 6& 12\end{array}\right)$ and state whether the linear system $P\mathbf{x}=\left(\begin{array}{c}5\\ 10\\ 15\end{array}\right)$ is consistent.

Step-by-step solution

1. Write the matrix $P$ and notice that each row is a multiple of the first row.
2. Since all rows are linearly dependent, the maximum number of independent rows is 1, so $\mathrm{rank}(P)=1$.
3. Form the augmented matrix $[P|b]$ with $b=\left(\begin{array}{c}5\\ 10\\ 15\end{array}\right)$.
4. Check if $b$ is a linear combination of the rows of $P$; it must be $k\times$(first row) for some $k$.
5. From the first entry $k=5$, but then the third entry would be $5\times 4=20$, which is not $15$.
6. Hence the augmented matrix has rank 2 while $\mathrm{rank}(P)=1$; the system has no solution.

Concept tested: Rank and consistency of a linear system

Published 11 September 2026 · How this solution was prepared and checked