Skip to main content
Mathematics LibreTexts

11.4: Elementary Matrices

  • Page ID
    65070
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    from urllib.request import urlretrieve
    urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
                'answercheck.py');

    NOTE: A detailed description of elementary matrices can be found here in the Beezer text Subsection EM 340-345 if you find the following confusing.

    There exist a cool set of matrices that can be used to implement Elementary Row Operations. Recall our elementary row operations include:

    1. Swap two rows
    2. Multiply a row by a constant (\(c\))
    3. Multiply a row by a constant (\(c\)) and add it to another row.

    You can create these elementary matrices by applying the desired elementary row operations to the identity matrix.

    If you multiply your matrix from the left using the elementary matrix, you will get the desired operation.

    For example, here is the elementary row operation to swap the first and second rows of a \(3 \times 3\) matrix:

    \[\begin{split}
    E_{12}=
    \left[
    \begin{matrix}
    0 & 1 & 0\\
    1 & 0 & 0 \\
    0 & 0 & 1
    \end{matrix}
    \right]
    \end{split} \nonumber \]

    import numpy as np
    import sympy as sym
    sym.init_printing(use_unicode=True)
    A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])
    sym.Matrix(A)
    # 'Run' this cell to see the output
    E1 = np.matrix([[0,1,0], [1,0,0], [0,0,1]])
    sym.Matrix(E1)
    # 'Run' this cell to see the output
    A1 = E1*A
    sym.Matrix(A1)
    # 'Run' this cell to see the output
    Do This

    Give a \(3 \times 3\) elementary matrix named E2 that swaps row 3 with row 1 and apply it to the \(A\) Matrix. Replace the matrix \(A\) with the new matrix.

    # Put your answer here.  
    from answercheck import checkanswer
    
    checkanswer.matrix(E2,'2c2d2e407389eabeb6d90894565c830f');
    Do This

    Give a \(3 \times 3\) elementary matrix named E3 that multiplies the first row by \(c=3\) and adds it to the third row. Apply the elementary matrix to the \(A\) matrix. Replace the matrix \(A\) with the new matrix.

    # Put your answer here.
    from answercheck import checkanswer
    
    checkanswer.matrix(E3,'55ae1f9eb21df00c59dad623b9471506');
    Do This

    Give a \(3 \times 3\) elementary matrix named E4 that multiplies the second row by a constant \(c=1/2\) applies this to matrix \(A\).

    # Put your answer here.  
    from answercheck import checkanswer
    
    checkanswer.matrix(E4,'3a5256840ef907a1b73ebba4471ac26d');

    If the above are correct then we can combine the three operators on the original matrix \(A\) as follows.

    A = np.matrix([[3, -3,9], [2, -2, 7], [-1, 2, -4]])
    
    sym.Matrix(E4*E3*E2*A)

    This page titled 11.4: Elementary Matrices is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

    • Was this article helpful?