Why is there no matrix whose row space and nullspace both contain (1,1,1)?

Step 1 of 2

Consider a matrix A such that:

\(A=\left(\begin{array}{l}
a_{1} \\
\cdots \\
a_{n}
\end{array}\right)\) where \(a_{i}\) is the \(i^{t h}\) row of A and \(u^{T}\) is a vector of the null space of A.

Then as \(u^{T}\) is a vector of the null space, hence \(A u^{T}=0 \Rightarrow\left(\begin{array}{c}
a_{1} u^{T} \\
\ldots \\
a_{n} u^{T}
\end{array}\right)=0\) where \(a_{i} u^{T}=0\) for all i.