Home page for accesible maths 1 Fields and Matrices 1.A Fields 1.C Inverse matrices and triangular matrices

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1.B Row operations and reduced echelon form

At this point, an inquisitive student (which I hope you are) should be asking various questions, such as: “Why should we care about fields? What is the purpose of these axioms”? To answer these questions, let’s recall some facts from MATH105.

Definition 1.3:

Firstly, choose a field $F$ , which is usually either $F={\mathbb{R}}$ or $F={\mathbb{C}}$ . Then, let $A\in\operatorname{M}_{{n\times m}}({F})$ be matrix with $n$ rows and $m$ columns, whose coefficients are in the set $F$ . Then an elementary row operation (e.r.o.) is defined to be one of the following three operations:

1

$R_{i}=r_{i}+\lambda r_{j}$ For $\lambda\in F$ , add $\lambda$ times row $j$ to row $i$ .
2

$R_{i}=\lambda r_{i}$ For $0\neq\lambda\in F$ , multiply row $i$ by $\lambda$ .
3

$R_{i}\leftrightarrow R_{j}$ Swap rows $i$ and $j$ .

We also define elementary column operation (e.c.o.) to be the same as above, except replacing the rows ( $R_{i}$ and $r_{i}$ ) with columns ( $C_{i}$ and $c_{i}$ ). For example, multiplying a column by a non-zero scalar is an e.c.o., but not an e.r.o.

Theorem 1.4.

For $F={\mathbb{R}}$ , any e.r.o. moves matrices from $\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ to $\operatorname{M}_{{n\times m}}({{\mathbb{R}}})$ .

Considering the second e.r.o., this theorem says that if you take any matrix with real coefficients, and multiply a row by a real number, then you end up with another matrix with real coefficients. That’s pretty obvious. Now consider the next two theorems, which are only slightly less obvious.

Theorem 1.5.

For $F={\mathbb{C}}$ , any e.r.o. moves matrices from $\operatorname{M}_{{n\times m}}({{\mathbb{C}}})$ to $\operatorname{M}_{{n\times m}}({{\mathbb{C}}})$ .

Theorem 1.6.

For $F={\mathbb{Q}}$ , any e.r.o. moves matrices from $\operatorname{M}_{{n\times m}}({{\mathbb{Q}}})$ to $\operatorname{M}_{{n\times m}}({{\mathbb{Q}}})$ .

These are three distinct theorems, and you are expected to know all three of them. But they somehow seem “the same”. Rather than proving each one separately, we will prove the following generalization. Since ${\mathbb{R}},{\mathbb{C}},$ and ${\mathbb{Q}}$ are all fields, we are proving all three of the above theorems simultaneously. In the proof we are only allowed to use the field axioms.

Theorem 1.7.

For any field $F$ , any e.r.o. moves matrices from $\operatorname{M}_{{n\times m}}({F})$ to $\operatorname{M}_{{n\times m}}({F})$ .

Proof.

Firstly, it is clear that the size of the matrix does not change. So we only need to check that the coefficients of the new matrix are still in $F$ .

Consider the first row operation $R_{i}=r_{i}+\lambda r_{j}$ , where $\lambda\in F$ . It changes the coefficients of the $i$ th row from $a_{ik}$ to $a_{ik}+\lambda a_{jk}$ . By F2, $\lambda a_{jk}\in F$ , and therefore by F1 we know $a_{ik}+\lambda a_{jk}\in F$ . This is true for every $k=1,2,\cdots,m$ , and so the new row has coefficients all in $F$ .

Next, consider the second row operation $R_{i}=\lambda r_{i}$ , where $0\neq\lambda\in F$ . Again, by F2, multiplication is a binary operation, so the new coefficients $\lambda a_{ik}$ are all still in $F$ .

Finally, for the third row operation, the coefficients after the swapping operation are still all in $F$ . ∎

Exercise 1.8:

For $F={\mathbb{R}}$ , give an example of a matrix in $\operatorname{M}_{{3}}({{\mathbb{Q}}})$ , and an e.r.o. (over ${\mathbb{R}}$ ) which moves your matrix to a matrix not in $\operatorname{M}_{{3}}({{\mathbb{Q}}})$ .

[End of Exercise]

We will say that the matrix $B$ is row-equivalent to the matrix $A$ if $B$ can be obtained from $A$ by performing a finite sequence of elementary row operations on $A$ (in fact, this forms an equivalence relation; see Exercise 1.19). The following definition and theorem should also be familiar from MATH105.

Definition 1.9:

A matrix is in reduced row echelon form if the following conditions are satisfied:

i.

The leading coefficient in each non-zero row is 1
ii.

Each leading coefficient is the only non-zero entry in its column
iii.

All the zero rows are in the bottom rows, and as the row numbers increase, the column numbers of the leading coefficients also (strictly) increase; i.e. the matrix is in echelon form.

Theorem 1.10.

Let $F$ be a field. Then any matrix with coefficients in $F$ can put into reduced echelon form by a sequence of e.r.o.’s. Furthermore, the reduced echelon form of any matrix is unique; in other words, it is independent of the sequence of e.r.o.’s.

The above theorem should be familiar when the field is $F={\mathbb{R}}$ . We will not give the proof; the general case of the proof is the same as the real case, because the only properties of the real numbers that were used are contained in the list of field axioms F1 to F11.

Exercise 1.11:

Are the matrices $\begin{bmatrix}1&2&0\\ 0&0&1\\ 0&0&0\end{bmatrix}$ and $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}\in\operatorname{M}_{{3}}({{\mathbb{R}}})$ row-equivalent to each other? Justify your answer by applying Theorem 1.10.

[End of Exercise]