Homework 12 - Determinants

1. Compute the determinant of $$\begin{pmatrix} 1 & -1 & 0 & 2 \\ -2 & 1 & 1 & -1 \\ 1 & 0 & -1 & 1 \\ 0 & 2 & -1 & -1 \end{pmatrix}$$ by hand using both Gaussian Elimination and a cofactor expansion. Feel free to check your answer using Sage.

2. A matrix $A$ is called nilpotent if there is some $l > 0$ with $A^l = 0$. What is the determinant of $A$?

   If $\det A = 0$, is $A$ necessarily nilpotent?

3. Let $A$ be an $m \times n$ matrix with $m < n$. Show that if the rows of $A$ are linearly dependent then the determinant of any $m \times m$ matrices obtained from $A$ by deleting $n-m$ columns must be $0$.

   Conclude that, if at least one of the $m \times m$ matrices obtained from $A$ by deleting $n-m$ columns from $A$ has nonzero determinant, then the rows of $A$ are linearly independent.

4. Let $B$ be a $m \times 2$ matrix. Assume all the determinants of the $2 \times 2$ matrices obtained from $B$ by deleting all but $2$ rows are $0$. Show that all rows of $B$ are scalar multiples of a single row.

   It is also true that, for $m > n$, if all $m \times m$ sub- matrices of $m \times n$ $B$ have zero determinant then the rows of $B$ are linearly dependent.

5. A matrix $n \times n$ is block diagonal if we have some $1 \leq n_i \leq n$ with with $\sum n_i = n$ and for each $1 \leq i \leq s$ there are $n_i \times n_i$ matrices $A_i$ and $$A = \begin{pmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_s \end{pmatrix}$$ So for example, $$\begin{pmatrix} 1 & 2 & 0 \\ 3 & 4 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$ is a block diagonal matrix with a $2 \times 2$ block and a $1 \times 1$ block.

   Show that $$\det A = \prod_{i=1}^s \det A_i$$