In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.
Carry out the following matrix calculations.
\[ \begin{pmatrix} 0 & 1 & 0 \end{pmatrix} \begin{pmatrix}a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\cr\end{pmatrix} \begin{pmatrix}1\cr0\cr0\cr\end{pmatrix}\] and
\[\begin{pmatrix} 0 & 1 & 0\end{pmatrix} \begin{pmatrix} a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\end{pmatrix} \begin{pmatrix}0\cr1\cr0\cr\end{pmatrix} \]
What matrix multiplication would you do if you wanted the answer to be \(a_{31}\)?
In the first question above, the bra/ket representations for the calulations are:
\[\left\langle {2}\right| A\left|{1}\right\rangle = ? \quad \hbox{and} \quad \left\langle {2}\right| A\left|{2}\right\rangle = ?\]
Write the second question in bra/ket notation.
Little introduction is needed.
Students need to be familiar with (1) matrix multiplication and (2) how to write the standard basis in Dirac notation.