Since 1 April 2011 I from time to time was trying to convince Wolfram Alpha to fix a bug in the way they computed eigenvectors, see my post of 28 April 2012. It survived until May 2016:

As you can see, Wolfram Alpha was thinking that the zero vector is eigenvector. On 5 May 2016 this bug was finally fixed:

But there is still one glitch which can send an undergraduate student on a wrong path. The use of round brackets as delimeters for both matrices and vectors suggests that the vector \((1,0)\) is treated as a \( 1 \times 2\) matrix, that is a **row vector**. This determines which way it can be multiplied by a \(2 \times 2 \) matrix: on the right, that way:

\[

(1,0) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)

\]

and not that way

\[

\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right)(1,0),

\]

the latter is simply not defined. Therefore the correct answer is not

\[

\mathbf{v}_1 = (1,0)

\]

but

\[ \mathbf{u} = (0,1) \quad\mbox{ or }\quad \mathbf{w} = (1,0)^T = \left(\begin{array}{c} 1 \\ 0\end{array}\right),

\]

depending on convention used for vectors:** row vectors** or **column vectors**. Indeed if

\[

A = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right),

\]

then

\[ \mathbf{v}_1A = (1,0)\left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (1,2) \ne 1\cdot \mathbf{v}_1,

\]

while

\[

A\mathbf{w} = \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) \left(\begin{array}{c} 1 \\ 0\end{array}\right) = \left(\begin{array}{c} 1 \\ 0\end{array}\right) = 1\cdot \mathbf{w}

\]

and

\[

\mathbf{u} A = (0,1) \left(\begin{array}{cc} 1 & 2 \\ 0 & 1\end{array}\right) = (0,1) = 1\cdot \textbf{u}.

\]

The bug is likely to sit somewhere in the module which converts matrices and vectors from their internal representation within the computational engine into the format for graphics output. It should be very easy to fix. It is not an issue of computer programming, it is just lack of attention to basic principle of exposition of mathematics and didactics of mathematics education.