Home page for accesible maths 2 Row operations on matrices 2.3 Rank of a matrix 3 Inverting matrices

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2.4. Exercises

Exercise 2.4.1.

For each of the following assertions decide whether it is true or not. Justify your answer.

(i)

$E_{12}(2)=E_{2}(2)$ .
(ii)

$E_{12}(\lambda)E_{2}(\mu)=E_{2}(\mu)E_{12}(\lambda\mu)$ for all non-zero $\lambda,\mu\in{\mathbb{R}}$ .
(iii)

$E_{24}E_{13}(\lambda)=E_{13}(\lambda)E_{24}$ for all non-zero $\lambda\in{\mathbb{R}}$ .
(iv)

$E_{2}(2)E_{24}(4)=E_{24}(4)E_{2}(2)$ .
(v)

$E_{2}(2)E_{24}(2)=E_{24}(4)E_{2}(2)$ .
(vi)

$E_{i}(2)E_{i}(\frac{1}{2})$ is the identity matrix for all indices $i$ .
(vii)

$E_{ij}^{2}$ is the identity matrix for all indices $i, j$ .
(viii)

$E_{ij}(\lambda)E_{ij}({\lambda}^{-1})$ is the identity matrix for all non-zero $\lambda\in{\mathbb{R}}$ and indices $i, j$ .

Exercise 2.4.2.

Consider matrices in $\operatorname{M}_{{3}}({{\mathbb{R}}})$ .

(i)
Write the elementary matrices corresponding to the following elementary row operations.
1. (a)
  
  $R_{2}\leftrightarrow R_{3}$ ,
2. (b)
  
  $R_{2}=2r_{2}$ ,
3. (c)
  
  $R_{1}=r_{1}-2r_{3}$ ,
(ii)

Write the above sequence of elementary row operations (starting with $R_{2}\leftrightarrow R_{3}$ ) as a product of elementary matrices. Hence, write the result when applying the above sequence of elementary operations on the identity matrix.
(iii)

Prove that

$E_{3}(2)E_{13}(-2)=E_{23}E_{12}(-1)E_{2}(2)E_{23}.$

Exercise 2.4.3.

Write the $3\times 3$ matrix $A=(a_{ij})$ with coefficients $a_{ij}$ defined by $a_{ij}=2+i-ij$ for all $1\leq i,j\leq 3$ . Find a sequence $L_{1},\dots,L_{k}$ of elementary matrices in $M_{3}({\mathbb{R}})$ such that $L_{k}\cdots L_{1}A$ is in reduced echelon form.

Exercise 2.4.4.

Find the reduced echelon form of the following matrices, and state their ranks.

(i)

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

$\begin{pmatrix}7&3&5&-1&1\\ 3&-1&4&0&2\\ 3&15&-9&-3&-3\end{pmatrix}$
(iii)

$\begin{pmatrix}-2&2&1&-1\\ 4&8&4&-8\\ 1&2&0&-1\\ 0&6&2&-4\end{pmatrix}$
(iv)

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

$\begin{pmatrix}7&-2\\ 2&1\\ 9&0\\ 4&4\end{pmatrix}$

Exercise 2.4.5.

In each of the following examples, find a sequence $L_{1},\dots,L_{k}$ of elementary matrices such that $L_{k}\cdots L_{1}A$ is in reduced echelon form. ( $k$ will probably be different in each case.)

(i)

$A=\begin{pmatrix}1&4\\ 4&1\end{pmatrix}$
(ii)

$A=\begin{pmatrix}2&-1&0&3\\ 3&1&-1&6\\ 1&0&0&0\end{pmatrix}$
(iii)

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

$A=\begin{pmatrix}5&-2\\ 3&1\\ -2&3\\ 0&1\end{pmatrix}$
(v)

$A=\begin{pmatrix}1&2&3\\ 4&5&6\\ 5&7&9\end{pmatrix}$

Exercise 2.4.6.

Prove the following statements; below $i, j,$ and $k$ are assumed to be positive integers, all different from each other, and $\mu,\lambda\in{\mathbb{R}}$ :

(i)

$E_{ij}E_{ik}=E_{jk}$ .
(ii)

$E_{ij}^{2}=I_{n}$ .
(iii)

$E_{i}(\lambda)E_{j}(\mu)=E_{j}(\mu)E_{i}(\lambda)$ .
(iv)

$E_{i}(\lambda)E_{i}(\mu)=E_{i}(\lambda\mu)$ .
(v)

$E_{ij}(\lambda)E_{ij}(\mu)=E_{ij}(\mu)E_{ij}(\lambda)=E_{ij}(\lambda\mu)$ .

Exercise 2.4.7.

(i)

If $A\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ is in reduced echelon form, is it always true that $A^{2}=A$ ? If so, give a proof; and if not, then give a counter-example.
(ii)

Let $A,B\in\operatorname{M}_{{n}}({{\mathbb{R}}})$ be matrices in reduced echelon form. Must $A\cdot B$ also be in reduced echelon form? If so, give a proof; if not, then give a counter-example.