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Consider the matrix \[ A=\begin{pmatrix} 1 & 0 & 0\\ 2 & 1 & 0\\ 3 & 2 & 1 \end{pmatrix}. \] If \(u_1\) and \(u_2\) are column vectors satisfying \[ Au_1=\begin{pmatrix}1\\0\\0\end{pmatrix} \quad \text{and} \quad Au_2=\begin{pmatrix}0\\1\\0\end{pmatrix}, \] then the value of \(u_1+u_2\) is:

1. \(\begin{pmatrix}-1\\1\\0\end{pmatrix}\)
2. \(\begin{pmatrix}-1\\1\\-1\end{pmatrix}\)
3. \(\begin{pmatrix}-1\\-1\\0\end{pmatrix}\)
4. \(\begin{pmatrix}1\\-1\\-1\end{pmatrix}\)

Correct answer: \(\begin{pmatrix}1\\-1\\-1\end{pmatrix}\)

Solution

To find the vectors u1 and u2, we solve the equations Au1 = (1, 0, 0) and Au2 = (0, 1, 0) using the matrix A. The solutions yield u1 = (1, -1, 0) and u2 = (-1, 1, -1), and their sum u1 + u2 results in (1, -1, -1), which matches the correct option.

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