Home page for accesible maths 3 Inverting matrices 3.3 Invertible matrices as a product of elementary matrices 4 Determinants

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3.4. Exercises

Exercise 3.4.1.

In each of the following examples, using the augmented matrix method, find the inverse of the given matrix $A$ , if it exists.

(i)

$\begin{pmatrix}1&2&3\\ 2&3&4\\ 3&4&6\end{pmatrix}$
(ii)

$\begin{pmatrix}1&3&1\\ 2&6&3\\ -1&2&5\end{pmatrix}$
(iii)

$\begin{pmatrix}1&2&0\\ 0&0&1\\ 1&-2&3\end{pmatrix}$
(iv)

$\begin{pmatrix}1&0&-1\\ 2&-2&0\\ 0&3&-3\end{pmatrix}$
(v)

$\begin{pmatrix}0&2&-3\\ 1&1&1\\ -1&2&0\end{pmatrix}$
(vi)

$\begin{pmatrix}1&2&4&0\\ 0&1&1&3\\ 2&1&0&1\\ 0&1&4&-2\end{pmatrix}$
(vii)

$\begin{pmatrix}1&0&-1&0\\ 2&1&-2&0\\ 0&0&3&-3\\ 1&2&1&2\end{pmatrix}$
(viii)

$\begin{pmatrix}3&-1&0&2&1\\ 2&-2&-2&3&0\\ 1&0&1&-1&0\\ 2&1&3&-2&1\\ 0&1&0&0&0\end{pmatrix}$
(ix)

$\begin{pmatrix}3&2&1&0&1\\ 0&1&2&0&1\\ 1&1&1&1&1\\ 2&4&3&4&2\\ 1&0&2&1&2\end{pmatrix}$

Exercise 3.4.2.

For each invertible matrix in the previous Exercise, find elementary matrices $L_{1},\dots,L_{k}$ such that ${A}^{-1}=L_{k}\cdots L_{1}$ , where $k$ is some integer. Pay close attention to the order of multiplication.

Exercise 3.4.3.

Let $b,c,d\in{\mathbb{R}}$ . Find

\begin{pmatrix}0&b\\ c&d\end{pmatrix}^{-1}\quad\hbox{if it exists.}\quad

Exercise 3.4.4 (True or false?).

For each of the following statements, decide whether it is true or false. Justify your answer, by supplying a proof, or a counter-example.

(i)

Let $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ . $A$ is invertible if and only if $\operatorname{rank}A=3$ .
(ii)

Let $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ . $A$ is the zero matrix if and only if $\operatorname{rank}A=0$ .
(iii)

For any matrices $A,B\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ , we have $\operatorname{rank}(A+B)=\operatorname{rank}A+\operatorname{rank}B$ .
(iv)

There exists a matrix $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ with $\operatorname{rank}A=2$ and $\operatorname{rank}A^{2}=2$ .
(v)

For any matrices $A,B\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ , we have $\operatorname{rank}(AB)=\min\{\operatorname{rank}A,\operatorname{rank}B\}$ .
(vi)

There exists a matrix $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ with $\operatorname{rank}A=2$ and $\operatorname{rank}A^{2}=1$ .
(vii)

There exists a matrix $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ with $\operatorname{rank}A=1$ and $\operatorname{rank}A^{2}=2$ .
(viii)

Assume $A,B\in\operatorname{M}_{{2}}({{\mathbb{R}}})$ , with $A$ invertible. Then $\operatorname{rank}(AB)=\operatorname{rank}B$ .

Exercise 3.4.5.

For a positive integer $m\geq 1$ , the set

\mathbb{Z}_{m}:=\{0,1,2,\cdots,m-2,m-1\}

is called the integers modulo $m$ (also called the congruence classes modulo $m$ ). Within this set, we can add and multiply elements. For example, when $m=12$ , we have $9+5=2\in\mathbb{Z}_{12}$ . (You could think “9am + 5 hours = 2pm”; this is also written $14\equiv 2\ \textrm{mod}\ 12$ ).

(i)

Verify that within $\operatorname{M}_{{2}}({\mathbb{Z}_{3}})$ we have $\begin{pmatrix}1&1\\ 2&1\end{pmatrix}\begin{pmatrix}2&1\\ 2&2\end{pmatrix}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ .
(ii)

There are 81 matrices within $\operatorname{M}_{{2}}({\mathbb{Z}_{3}})$ . How many of them are invertible? In other words, for which matrices $A\in\operatorname{M}_{{2}}({\mathbb{Z}_{3}})$ does there exist a $B\in\operatorname{M}_{{2}}({\mathbb{Z}_{3}})$ such that $AB=I_{2}$ ?
[Hint: Of the 16 matrices in $\operatorname{M}_{{2}}({\mathbb{Z}_{2}})$ , 6 of them are invertible.]