Small Group Activity: Finding Matrix Elements

Quantum Fundamentals 2022
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.
What students learn Matrix multiplication with standard basis vectors. How to pull out a single matrix element by successive matrix multiplications. Matrix elements are bra/matrix/kets in Dirac notation. There is a relationship between standard basis elements with the rows and columns of a matrix.
  1. 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

    \[\begin{pmatrix} 0 & 1 & 0\end{pmatrix} \begin{pmatrix} a_{11} & a_{12} & a_{13} \cr a_{21} & a_{22} & a_{23} \cr a_{31} & a_{32} & a_{33}\end{pmatrix} \begin{pmatrix}0\cr1\cr0\cr\end{pmatrix} \]

  2. What matrix multiplication would you do if you wanted the answer to be \(a_{31}\)?

  3. 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.

Student Conversations

  • Students need help seeing how multiplying by the standard basis vectors picks off rows or columns. Students who think about overlaying columns/rows on matrices during matrix multiplication seem to have an easier time.

Author Information
Corinne Manogue, Kerry Browne, Maggie Greenwood
matrix multiplication dirac notation
Learning Outcomes