Exercises

01

Nilpotent matrix

Define the matrix $A_{\lambda }$:

$$A_{\lambda }=\left(\begin{array}{ccccc}0& \lambda & 0& 0& 0\\ 0& 0& \lambda & 0& 0\\ 0& 0& 0& \lambda & 0\\ 0& 0& 0& 0& \lambda \\ 0& 0& 0& 0& 0\end{array}\right).$$

For $\lambda =2$, calculate the powers $A_2$, $A_2^2$, $A_2^3$, $A_2^4$, $A_2^5$.

(What special property does $A_{\lambda }$ have? A matrix with that property is said to be nilpotent.)

02

Reflections square to identity

Show that ${\left[{\mathrm{refl}}_{\mathbf{v}}\right]}^2=I_2$ for any $\mathbf{v}=\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)$ with $\mathbf{v}\cdot \mathbf{v}=1$.

This means that reflecting twice across the same line gives back the original vector. (In 2D the hyperplane of reflection is just a line.)

03

Checking formula for $2\times 2$ inverse

Check the formula for $A^{-1}$ for a $2\times 2$ matrix $A$ by computing by hand $A{A}^{-1}$ and $A^{-1}A$. (You should obtain $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ in both cases.)

Which $2\times 2$ matrices $A$ have an inverse?

(Hint: you just checked a formula that works whenever it makes sense, so there is an inverse whenever it makes sense, given by the formula. Suppose $a,b,c,d$ are such that the formula does not make sense. Can you find a vector $\mathbf{u}\ne 0$ which is sent to $0$ by $A$?)

04

Inverse of a reflection

Show (in 2D) that the matrix of a reflection is the inverse of itself. Show this first abstractly (using the definition of inverse), and then concretely (using the formulas for reflection and inverse matrices).

05

Inverse of a rotation

Show (in 2D) that the inverse of the matrix of rotation by $\theta$ is the matrix of rotation by $-\theta$.

06

Inverse matrix to find preimage

Repeat the previous example for this data:

$$A=\left(\begin{array}{cc}1& -3\\ -2& 4\end{array}\right),\mathbf{v}=\left(\begin{array}{c}-4\\ 2\end{array}\right).$$