Prove the existence of row-reduced matrices with restrictions

In summary: But the problem is exactly on prove the first statement (ad = bc implies that a+ b = k*(d+c)). How can i state this in a mathematical manner?Analysing the first question I found an inconsistency. The matrix could be, say:\left(\begin{array}{cc} 1 & b\\ 0 & 0 \end{array}\right)Is that a problem? because from the point of view of matrices, these two are different.oh thanx...
  • #1
rockerman
14
0
Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices...



so i realize that there are seven possible 2x2 matrices that are row-reduced.
[1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1], [1 0; 0 0], [0 1; 0 0], [0 0; 0 0]...
and the only one that satisfies the restriction is the last. What am i missing? thx


This is the problem 6 of the section 1.3 of HOFFMAN and KUNZE - Linear Algebra.
 
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  • #2
Hi rockerman! :smile:

How did you define a row-reduced matrix?

I ask this because I highly doubt that

1) [1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1], [1 0; 0 0], [0 1; 0 0], [0 0; 0 0] are the only row-reduced matrices. In fact, I think there are infinitely many of them.

2) I don't think [0 1; 1 0], [0 0; 1 0], [0 0;0 1] even are row-reduced.

But all of this depends on the definition you're using. So, what is it?
 
  • #3
so..

the row reduced definition provided by the book is given as follows:

An mxn matrix R is called row-reduced if:

a) the first non-zero entry in each non-zero row of R is equal to 1;
b) each column of R which contains the leading non zero entry of some row has all its other entries 0.

This way, i think that only exists seven row-reduced matrices in R^(2x2).. but I'm not figuring out how to prove the existence of three such matrices..
 
  • #4
What about matrices like

[tex]\left(\begin{array}{cc} 1 & a\\ 0 & 0 \end{array}\right)[/tex]

Isn't that row-reduced too?
 
  • #5
oh, how I've missed this...


so there are many row-reduced matrices like that... but how i will find the three matrices that satisfies the condition?
 
  • #6
Choose a special a...
 
  • #7
the a that satisfies the condition is a= -b -c -d. So the three possible matrices are:

[tex]\left(\begin{array}{cc} 1 & a\\ 0 & 0 \end{array}\right)[/tex], [tex]\left(\begin{array}{cc} 0 & 0\\ 1 & a \end{array}\right)[/tex] and [tex]\left(\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right)[/tex]

is that correct?
 
  • #9
I was limiting my thinking to the most obvious cases of existence of such matrices. many thanks for the help; By now i have only one more doubt about my current exercise set. can i post on this topic?

I'll better read on the forum rules;
 
  • #10
I don't think the rules say anything about such a things. So you can post it here if you want :smile:
 
  • #11
oh thanx...
The problem
Consider the system of equations AX = 0 where
[tex]A = \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)[/tex]
is a 2x2 matrix over the field F. Prove the following

(a) If every entry of A is 0 then every pair (x1, x2) is a solution of AX = 0;
(b) if ad-bc != 0, the system AX = 0 has only the trivial solution x1 = x2 =0;

Attempt to solution:
(a) is pretty straightforward. I just show the form of the system {0*x1 + 0*x2 = 0; 0*x1 + 0*x2 = 0
and by the real numbers properties i conclude that any (x1, x2) satisfies the equation.

(b) let me in doubt. So far in the book, the word determinant even is mentioned. So i can't use this as a reasonable argument. I claim the follow:

if ad = bc then i can write a + b = k*(d+ c). By this way the process to make a row-reduced diagonal matrix will fail, resulting in other non-zero solutions. But the problem is exactly on prove the first statement (ad = bc implies that a+ b = k*(d+c)). How can i state this in a mathematical manner?
 
Last edited:
  • #12
Analysing the first question I found an inconsistency. The matrix could be, say:

[tex]\left(\begin{array}{cc} 1 & b\\ 0 & 0 \end{array}\right)[/tex]

Is that a problem? because from the point of view of matrices, these two are different.
 
  • #13
rockerman said:
oh thanx...
The problem
Consider the system of equations AX = 0 where
[tex]A = \left(\begin{array}{cc} a & b\\ c & d \end{array}\right)[/tex]
is a 2x2 matrix over the field F. Prove the following

(a) If every entry of A is 0 then every pair (x1, x2) is a solution of AX = 0;
(b) if ad-bc != 0, the system AX = 0 has only the trivial solution x1 = x2 =0;

Attempt to solution:
(a) is pretty straightforward. I just show the form of the system {0*x1 + 0*x2 = 0; 0*x1 + 0*x2 = 0
and by the real numbers properties i conclude that any (x1, x2) satisfies the equation.

(b) let me in doubt. So far in the book, the word determinant even is mentioned. So i can't use this as a reasonable argument. I claim the follow:

if ad = bc then i can write a + b = k*(d+ c). By this way the process to make a row-reduced diagonal matrix will fail, resulting in other non-zero solutions. But the problem is exactly on prove the first statement (ad = bc implies that a+ b = k*(d+c)). How can i state this in a mathematical manner?

Well, for (b), can't you just solve the system of equations by row reducing things? Just try to reduce the matrix and you'll see where ad-bc comes in!
 
  • #14
rockerman said:
An mxn matrix R is called row-reduced if:

a) the first non-zero entry in each non-zero row of R is equal to 1;
b) each column of R which contains the leading non zero entry of some row has all its other entries 0.
Oh weird. I would have expected "row-reduced" to include that the ones are moved to the top and sorted; e.g.
[tex]\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} \right)[/tex]
is row reduced, but neither of the following are:
[tex]\left( \begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right)[/tex]
[tex]\left( \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right)[/tex]
 
  • #15
Hurkyl said:
Oh weird. I would have expected "row-reduced" to include that the ones are moved to the top and sorted; e.g.
[tex]\left( \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix} \right)[/tex]
is row reduced, but neither of the following are:
[tex]\left( \begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} \right)[/tex]
[tex]\left( \begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} \right)[/tex]

as micromass stated, it is has to do with the definition that you are using... in this case the book (HOFFMAN - Linear Algebra) states as being the rows at any order.
 
  • #16
great tip micromass! thanx...

I don't know, but maybe i just have to look at the problem at the simplest manner possible...

the first thing that i did before was:
[tex](1) -> A = \left( \begin{matrix} 1 & b/a \\ c & d \end{matrix} \right)[/tex]
by doing so, i was stuck and nothing could be done anymore.
 

Related to Prove the existence of row-reduced matrices with restrictions

What is a row-reduced matrix?

A row-reduced matrix is a special type of matrix where the entries are manipulated through elementary row operations to simplify the matrix and make it easier to work with. The goal of row-reduction is to transform the matrix into a simpler form without changing its essential properties.

What restrictions are placed on row-reduced matrices?

The main restriction on row-reduced matrices is that they must be in row-echelon form. This means that the leading entry (the first non-zero entry) of each row must be to the right of the leading entry of the row above it. Additionally, all entries below the leading entry in each row must be zero.

Why is it important to prove the existence of row-reduced matrices with restrictions?

Proving the existence of row-reduced matrices with restrictions is important because it helps us understand the properties of these matrices and how they can be used in various applications. It also allows us to develop efficient algorithms for manipulating and solving systems of linear equations.

What are some applications of row-reduced matrices?

Row-reduced matrices are commonly used in solving systems of linear equations, finding the inverse of a matrix, and performing Gaussian elimination. They are also used in computer graphics, statistics, and other fields where matrix operations are necessary.

Can any matrix be row-reduced?

Yes, any matrix can be row-reduced. However, not all matrices can be row-reduced to a unique form. Some matrices may have multiple row-reduced forms, while others may not be able to be row-reduced at all, depending on the restrictions and properties of the matrix.

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