Zero determinant - Can we make a zero column?

In summary: Regarding the puzzle - good luck :). It can be quite conceptually challenging. There are no cheap shots involved, I can assure you of that. There is no friction you can blame, either ;). The first question is easier, and I'll give you a hint - look up the full statement of the theorem of conservation of (mechanical) energy. The second part is quite more difficult, conceptually speaking. Happy riddling :).
  • #1
Himanshu
67
0
I just wanted to know that the following statement is always true or not.

After I expand the determinant I get the value of the determinant as zero, ie. I know that the value of the determinant zero.

Then with the help of row or column transformations can we transform the determinant into another one that contains at least one row or one column, all whose elements are zero.
 
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  • #2
Himanshu said:
Then with the help of row or column transformations can we transform the determinant into another one that contains at least one row or one column, all whose elements are zero.

Row and column transformations apply to matrices, not determinants.
However, if you substitute "matrix" for "determinant" in the above quote, it is true. It basically means that your (square) matrix represents a degenerate set of linear equations, i.e., one that has an infinite number of solutions.


Assaf.
http://www.physicallyincorrect.com" [Broken]
 
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  • #3
ozymandias said:
Row and column transformations apply to matrices, not determinants.

The Row and column transformations that I was talking about is of determinant. I think I should rephrase "Row and column transformations" as "Row and column operations".
 
  • #4
My apologies, slight terminology misunderstanding on my part :).
We are talking about the same thing. I was thinking more in the direction of elementary operations used in Gaussian elimination, which are more or less the same thing you're talking about.
The answer remains: yes, you are correct. In fact, you can even make a stronger statement (if-and-only-if).

Assaf.
http://www.physicallyincorrect.com/" [Broken]
 
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  • #5
Yes. I thought the same. I have tried it on few examples. But how do I prove it in general. I mean if it is a theorem there must be a proof for it.

By the way I was going through the An Energy Conservation Puzzle on your website. It's really whacking my brain out. Is the solution very simple(ie. does it require only brainwork).
 
  • #6
Hey Himanshu,

There are proofs, of course, but as with anything in mathematics, how easy they are depends on what you assume you know.
Proving zero-column-->det(A)=0 is easy, since we can make that column the first, and hence all of the terms in the determinant will have some element in the row (a zero) multiplying them.
The converse, det(A)=0 --> zero-column, is a bit trickier. A heuristic argument (but not a proof) relies on a theorem stating that a square matrix A is invertible iff det(A) is non-zero. This means that, if det(A)=0, A is non-invertible, so we don't have a unique solution to A*v=b (had A been invertible, the solution would've been v=A^(-1)*b). This corresponds, by Gaussian elimination, to a row or column having all-zeros.
I'm afraid that for a full, rigorous proof you'll have to consult a linear algebra textbook.

Regarding the puzzle - good luck :). It can be quite conceptually challenging. There are no cheap shots involved, I can assure you of that. There is no friction you can blame, either ;). The first question is easier, and I'll give you a hint - look up the full statement of the theorem of conservation of (mechanical) energy.
The second part is quite more difficult, conceptually speaking.
Happy riddling :).

Assaf.
http://www.physicallyincorrect.com/" [Broken]
 
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1. What is a zero determinant?

A determinant is a mathematical value that represents the properties of a square matrix. A zero determinant means that the matrix does not have an inverse, and therefore it cannot be inverted or solved for.

2. Why is it important to avoid having a zero column in a matrix?

Having a zero column in a matrix means that the matrix is singular, which means it has no inverse. This can lead to errors in calculations and make the matrix unsolvable, making it difficult to use in many scientific and engineering applications.

3. Can we make a zero column in a matrix?

Yes, it is possible to create a zero column in a matrix by performing certain operations, such as adding or subtracting a multiple of one column to another. However, it is important to note that this will result in a zero determinant and make the matrix unsolvable.

4. What can we do if we accidentally create a zero column in a matrix?

If a zero column is accidentally created in a matrix, the best solution is to try and avoid using that matrix in calculations. If it is necessary to use the matrix, alternative methods such as using a pseudoinverse or approximating the values may be used.

5. How can we prevent having a zero column in a matrix?

To prevent having a zero column in a matrix, it is important to carefully perform operations on the matrix and avoid adding or subtracting a multiple of one column to another. Additionally, using different mathematical techniques or algorithms may help to avoid creating a zero column.

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