7.6: Review Problems

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.

Contributor

• David Cherney, Tom Denton, and Andrew Waldron (UC Davis)