Activity3.4.4

We will explore cofactor expansions through some examples.

  1. Using a cofactor expansion, show that the determinant of the following matrix

    \begin{equation*} \det \left[\begin{array}{rrr} 2 \amp 0 \amp -1 \\ 3 \amp 1 \amp 2 \\ -2 \amp 4 \amp -3 \\ \end{array}\right] = -36 \text{.} \end{equation*}

    Remember that you can choose any row or column to create the expansion, but the choice of a particular row or column may simplify the computation.

  2. Use a cofactor expansion to find the determinant of

    \begin{equation*} \left[\begin{array}{rrrr} -3 \amp 0 \amp 0 \amp 0 \\ 4 \amp 1 \amp 0 \amp 0 \\ -1 \amp 4 \amp -4 \amp 0\\ 0 \amp 3 \amp 2 \amp 3 \\ \end{array}\right] \text{.} \end{equation*}

    Explain how the cofactor expansion technique shows that the determinant of a triangular matrix is equal to the product of its diagonal entries.

  3. Use a cofactor expansion to determine whether the following vectors form a basis of \(\real^3\text{:}\)

    \begin{equation*} \threevec{2}{-1}{-2}, \threevec{1}{-1}{2}, \threevec{1}{0}{-4} \text{.} \end{equation*}
  4. Sage will compute the determinant of a matrix A with the command A.det(). Use Sage to find the determinant of the matrix

    \begin{equation*} \left[\begin{array}{rrrr} 2 \amp 1 \amp -2 \amp -3 \\ 3 \amp 0 \amp -1 \amp -2 \\ -3 \amp 4 \amp 1 \amp 2\\ 1 \amp 3 \amp 3 \amp -1 \\ \end{array}\right] \text{.} \end{equation*}

in-context