# Harrison Chapman

## Linear Algebra I: Homework 3

Due: Friday, September 8, 2017
1. Find the inverse $A^{-1}$ of the matrix $A$:

We can solve for $A^{-1}$ using Gaussian elimination on the augmented matrix $(A \mid I)$ to get $(I \mid A^{-1})$:

So,

2. Let $\mathbf{0}$ be the $2 \times 2$ matrix with all zero entries.

1. Is there a matrix $A \ne \mathbf 0$ for which $AA = \mathbf 0$? Justify your answer.

Yes, there are many examples. These matrices are called nilpotent matrices. For example,

1. Is there a matrix $B \ne \mathbf 0$ and $B \ne I$ for which $BB = B$? Justify your answer.

Yes, there are many examples. These matrices are called idempotent matrices. For example,

3. Find the inverse $R_\theta^{-1}$ of the matrix $R_\theta$:

$R_\theta$ is a $2 \times 2$ matrix, so we can just use the formula for the inverse.