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juantheron

Member

Registered: 2011-10-19

Posts: 312

non singular matrices

The no. of 3×3 non - singular matrices matrices, with four entries as 1 and all other entries as 0

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: non singular matrices

hi juantheron

To be non-singular you cannot have a row or column of all zeros.

So three 1s must be distributed 1 per column, 1 per row.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

juantheron

Member

Registered: 2011-10-19

Posts: 312

Re: non singular matrices

Thanks Bob

I want more explanation

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Hi;

By direct count I get 36:

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: non singular matrices

hi juantheron

I don't have an answer but here's my thinking.

I've illustrated with three matrices below.

(i)  If you have a row ( or column) of zeros then the determinant is zero so the matrix is singular.

(ii) Assign three 1s so that every row and column has a single 1.  How many ways can you do this?

(iii) Fill one more space with the fourth 1,  How many ways can you do this?

(iv) Now allow for repeats caused by (iii)

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Hi juan

Maybe you could try getting the number of matrices which have one row of all 0 an then subtract from the total number of matrices.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

juantheron

Member

Registered: 2011-10-19

Posts: 312

Re: non singular matrices

Thanks Bob I am trying to understant it

To bobbym  How can we get it

To anonimnystefy how can we find total no. of matrices using 4 (1,s) and rest all zero

Thanks

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

For the first one you have 9 possibilities,for the second 8 possibilities,for the third one 7, and for the fourth 6. Multiply that and divide by 4! because we can permutate the 1s.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Hi juantheron;

I believe that is the correct answer but I have not found a way to count them using combinatorics. I will post as soon as I do.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Hi bobbym

Do you think it can be done through the negative event?

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

I do not know the answer to that. The ones that are singular do not follow a recognizable pattern.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Well,we find all the possible matrices with 4 ones and rest zeros. We subtract from that all the matrices with one row being all zeros. We thus get what we want.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

That one is singular and it has no row of all zeroes.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Sorry.Row/column.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Hi;

No row or column of zeroes but it is singular.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Can we use Kramer?

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

It will not help. There is just no rule for the singular matrices. How the ones should be placed that makes this countable.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

I thought we could just use the equation:

aei+bfg+cdh-afh-bdi-ceg=0

And then just put in 3 zeros instead of three letters. Then we just put in other zeros and ones. I think that is countable.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Those variables are more than that. They represent elements of the matrix. Only certain ones can have 0's or 1's. That is what we can not do, decide which ones are which.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Would it help to notice that each of those non-singular matrices have exactly one "Kramer diagonal" filled with zeros?

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Hi anonimnystefy;

Explain the kramer diagonal in relation to a nonsingular matrix.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Only one Kramer diagonal is filled up totally with zeros.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

Is there some theorem that says that a non-singular matrix has only one Kramer diagonal that is filled with zeros?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: non singular matrices

Oops,I have found one with no diagonals that fits our assumptions.

---

“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: non singular matrices

I can not find any specific matrix property that is clear enough to count by combinatorics. I am afraid juan is going to have stick with the answer I gave by direct count.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.