Activity 3.4.4

We will explore cofactor expansions through some examples.

1. Using a cofactor expansion, show that the determinant of the following matrix $$\det \left[\begin{array}{rrr} 2 & 0 & -1 \\ 3 & 1 & 2 \\ -2 & 4 & -3 \\ \end{array}\right] = -36$$. 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 $$\left[\begin{array}{rrrr} -3 & 0 & 0 & 0 \\ 4 & 1 & 0 & 0 \\ -1 & 4 & -4 & 0\\ 0 & 3 & 2 & 3 \\ \end{array}\right]$$. 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 $\real^3$: $$\threevec{2}{-1}{-2}, \threevec{1}{-1}{2}, \threevec{1}{0}{-4}$$.

4. Sage will compute the determinant of a matrix A with the command A.det(). Use Sage to find the determinant of the matrix $$\left[\begin{array}{rrrr} 2 & 1 & -2 & -3 \\ 3 & 0 & -1 & -2 \\ -3 & 4 & 1 & 2\\ 1 & 3 & 3 & -1 \\ \end{array}\right]$$.