Matrix Commutivity

[ChaoticLlama]

Let A be a 2x2 Matrix.
If A commutes with \(\displaystyle \left[ {\matrix{
0 & 1 \cr
0 & 0 \cr
} } \right]\), show that \(\displaystyle A = \left[ {\matrix{
a & b \cr
0 & a \cr

} } \right]\) for some a & b.

 

[pka]

\(\displaystyle \L
\begin{array}{rcl}
\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array}} \right]\left[ {\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}} \right] & = & \left[ {\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}} \right]\left[ {\begin{array}{cc}
a & b \\
c & d \\
\end{array}} \right] \\
\left[ {\begin{array}{cc}
0 & a \\
0 & c \\
\end{array}} \right] & = & \left[ {\begin{array}{cc}
c & d \\
0 & 0 \\
\end{array}} \right] \\
& \Rightarrow & a = d,\quad c = 0,\;\& \;b = \mbox{any number} \\
\end{array}\)

 

[ChaoticLlama]

thanks