$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 7.6: Review Problems

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

1. Find formulas for the inverses of the following matrices, when they are not singular:
a) $$\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix}$$
b) $$\begin{pmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \\ \end{pmatrix}$$

When are these matrices singular?

2. Write down all $$2\times 2$$ bit matrices and decide which of them are singular. For those which are not singular, pair them with their inverse.

3. Let $$M$$ be a square matrix. Explain why the following statements are equivalent:
a) $$MX=V$$ has a $$\textit{unique}$$ solution for every column vector $$V$$.

b) $$M$$ is non-singular.

Hint: In general for problems like this, think about the key words:

First, suppose that there is some column vector $$V$$ such that the equation $$MX=V$$ has two distinct solutions. Show that $$M$$ must be singular; that is, show that $$M$$ can have no inverse.

Next, suppose that there is some column vector $$V$$ such that the equation $$MX=V$$ has no solutions. Show that $$M$$ must be singular.

Finally, suppose that $$M$$ is non-singular. Show that no matter what the column vector $$V$$ is, there is a unique solution to $$MX=V.$$

4. Left and Right Inverses: So far we have only talked about inverses of square matrices. This problem will explore the notion of
a left and right inverse for a matrix that is not square. Let
$$A=\begin{pmatrix}0 & 1 & 1 \\ 1&1&0\end{pmatrix}$$

a) Compute:
(i) $$A A^{T}$$,
(ii) $$\big(A A^{T}\big)^{-1}$$,
(iii) $$B:=A^{T} \big(A A^{T}\big)^{-1}$$

b) Show that the matrix $$B$$ above is a $$\textit{right inverse}$$ for $$A$$, $$\textit{i.e.}$$, verify that
$$AB=I\, .$$
c) Does $$BA$$ make sense? (Why not?)

d) Let $$A$$ be an $$n\times m$$ matrix with $$n>m$$. Suggest a formula for a left inverse $$C$$ such that
$$CA=I$$
$$\textit{Hint: you may assume that A^TA has an inverse.}$$

e) Test your proposal for a left inverse for the simple example
$$A=\begin{pmatrix}1\\2\end{pmatrix}\, ,$$
f) True or false: Left and right inverses are unique. If false give a counterexample.

5. Show that if the range (remember that the range of a function is the set of all its possible outputs) of a $$3\times3$$ matrix $$M$$ (viewed as a function $$\mathbb{R}^{3}\to \mathbb{R}^{3}$$) is a plane then one of the columns is a sum of multiples of the other columns. Show that this relationship is preserved under EROs. Show, further, that the solutions to $$Mx=0$$ describe this relationship between the columns.

6. If $$M$$ and $$N$$ are square matrices of the same size such that $$M^{-1}$$ exists and $$N^{-1}$$ does not exist, does $$(MN)^{-1}$$ exist?

7. If $$M$$ is a square matrix which is not invertible, is $$\exp{M}$$ invertible?

8. Elementary Column Operations (ECOs) can be defined in the same 3 types as EROs. Describe the 3 kinds of ECOs. Show that if maximal elimination using ECOs is performed on a square matrix and a column of zeros is obtained then that matrix is not invertible.