Worksheet 23

1. Compute the determinant of $$\begin{pmatrix} 1 & -1 & 0 \\ 2 & 1 & 1 & \\ -1 & 1 & 3 \end{pmatrix}$$ by hand using a cofactor expansion.

2. Show that if $A$ is an $n \times n$ matrix with each $A_{ij} \in \mathbb{Z}$ then $\det A \in \mathbb{Z}$.

3. Show that the cofactor expansion satisfies $$\det(S(i,c)A) = c \det A$$ Try $n=2$ and $n=3$ to get a feel for the question.