How can I prove that A=0 using elementary operations?

In summary, the conversation discusses a homework problem involving proving that A=0 given certain properties of a matrix An*n. The conversation provides a hint to use elementary matrices and their properties to develop a proof for this problem.
  • #1
drosales
7
0
I need help with another homework problem

Let n be a positive integer and An*n a matrix such that det(A+B)=det(B) for all Bn*n. Show that A=0

Hint: prove property continues to hold if A is modified by any finite number of row or column elementary operations

It seems obvious that A=0 but I'm having trouble developing the proof. Any help would be great.
 
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  • #2
Please post homework in the homework forum. I moved it for you now.

A hint for the proof: can you write a row/column operation as an elementary matrix??
 
  • #3
Yes and the product of the elementary matrices returns
A=E1*E2*..*En

is this what you are referring to?
 
  • #4
Yes. Let E be an elementary matrix, can you show that

[tex]det(EA+B)=det(B)[/tex]

??
 
  • #5
Im not quite sure how to show this
 
  • #6
Hint: [itex]B=EE^{-1}B[/itex].

Use that [itex]det(XY)=det(X)det(Y)[/itex].
 

FAQ: How can I prove that A=0 using elementary operations?

What is a determinant?

A determinant is a numerical value that is calculated from a square matrix. It provides information about the properties of the matrix, such as whether it is invertible or singular.

How is a determinant calculated?

The determinant of a 2x2 matrix can be calculated by multiplying the elements on the main diagonal and subtracting the product of the elements on the other diagonal. For larger matrices, there are various methods such as cofactor expansion and Gaussian elimination to calculate the determinant.

What does the determinant tell us about a matrix?

The determinant provides information about the properties of a matrix, such as its invertibility, rank, and volume of the parallelepiped formed by the column vectors. It also plays a crucial role in solving systems of linear equations and finding the inverse of a matrix.

What is the significance of a determinant in linear algebra?

The determinant is a fundamental concept in linear algebra and is used in various applications, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and determining the invertibility of a matrix. It also has connections to other areas of mathematics, such as geometry and calculus.

Can the determinant be negative?

Yes, the determinant can be negative, positive, or zero. The sign of the determinant depends on the orientation of the coordinate system and the order of the elements in the matrix. A negative determinant indicates that the transformation results in a reflection or a rotation in addition to a change in scale.

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