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Matrix Calculations: An introduction to parts of linear algebra

Chrissy Safranski

Worksheet Participate
A4US

1. Why rref?

Write the associated system of linear equations for each of the matrices below.
\(\begin{bmatrix} 1 \amp 2 \amp 3 \amp 4 \\ 2 \amp -1 \amp 3 \amp 0 \\ 3 \amp 1 \amp 5 \amp 3 \end{bmatrix}\)
\(\begin{bmatrix} 1 \amp 0 \amp 0 \amp 4 \\ 0 \amp 1 \amp 0 \amp 0 \\ 0 \amp 0 \amp 1 \amp 3 \end{bmatrix}\)
Which system of equations would you prefer to be asked to solve on an exam?

2. Gauss-Jordan algorithm.

Use Gauss-Jordan elimination to put each of the matrices below into reduced row echelon form.
  1. \begin{equation*} \begin{bmatrix} 2 \amp 5 \amp -3 \\ 5 \amp -5 \amp -25 \end{bmatrix} \end{equation*}
  2. \begin{equation*} \begin{bmatrix} 2 \amp 5 \amp -3 \\ 2 \amp 5 \amp 1 \end{bmatrix} \end{equation*}
  3. \begin{equation*} \begin{bmatrix} 1\amp 2 \amp 3 \amp 5 \\ 0 \amp 4 \amp 5 \amp 7 \\ 1 \amp 6 \amp 9 \amp 11 \end{bmatrix} \end{equation*}

3. Form of matrices.

For each of the following matrices, say if it is in reduced row echelon form. If it isn’t, say whether it is in row echelon form or if it’s in neither form.
  1. \(\displaystyle \left[\begin{array}{ccc} 1\amp 0 \amp -3\\0\amp 1 \amp 1\\ \end{array} \right]\)
  2. \(\displaystyle \left[\begin{array}{cc} 1\amp 2\\0\amp 0\\ \end{array} \right]\)
  3. \(\displaystyle \left[\begin{array}{ccc} 1\amp 1 \amp -3\\0\amp 0 \amp 1\\ \end{array} \right]\)
  4. \(\displaystyle \begin{bmatrix} 1\amp 0 \amp -3\\0\amp 0 \amp 0\\ 0 \amp 1 \amp 3 \end{bmatrix}\)
  5. \(\displaystyle \begin{bmatrix} 1\amp 0 \amp 0 \amp 4 \amp 5 \\0\amp 0 \amp 1 \amp 2 \amp 7 \\ 0 \amp 1 \amp 0 \amp 3 \amp 11 \end{bmatrix}\)
  6. \(\displaystyle \begin{bmatrix} 1\amp 0 \amp 0 \amp 5 \\0\amp 4 \amp 0 \amp 7 \\ 0 \amp 0 \amp 1 \amp 11 \end{bmatrix}\)
  7. \(\displaystyle \begin{bmatrix} 1\amp 0 \amp 0 \amp 5 \\0\amp 1 \amp 0 \amp 7 \\ 0 \amp 0 \amp 1 \amp 11 \end{bmatrix}\)
  8. \(\displaystyle \begin{bmatrix} 1\amp 0 \amp 3 \amp 0 \amp 6 \\0\amp 1 \amp -1 \amp 0 \amp -23 \\ 0 \amp 0 \amp 0 \amp 1 \amp 11 \end{bmatrix}\)

4.

Write down the associated linear system of equations for Item 3.c, Item 3.g and Item 3.h. What observations and questions do you have about these systems?
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