Proving Determinant of NxN Matrix All Rows & Cols Sum to 0

In summary, the conversation discusses how to argue that all (n-1)*(n-1) minors of a given n*n matrix, where all rows and columns sum to 0, have the same determinant up to a sign. The conclusion is that this is likely achieved through the use of cofactor expansion and induction.
  • #1
robforsub
16
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If given a n*n matrix with all rows and columns sum to 0, how do I argue that all its (n-1)*(n-1) minor have the same determinant up to a sign?
Since all rows and columns all sum to 0, then I know that any column is a linear combination of all others, so that the determinant of this n*n matrix must be zero, then since the determinant is calculated using minors, it seems to imply that all (n-1)*(n-1) minors must have the same determinant up to a sign, but how do I rigorously prove that?
 
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  • #2
This is just a hunch, but cofactor expansion and induction are probably involved.
 

1. What is the determinant of a matrix?

The determinant of a matrix is a numerical value that represents certain properties of the matrix, including its size, shape, and orientation.

2. How is the determinant of a matrix calculated?

The determinant of a matrix can be calculated by using various methods, such as the cofactor expansion method or the Gaussian elimination method.

3. What does it mean if all rows and columns of an NxN matrix sum to 0?

If all rows and columns of an NxN matrix sum to 0, it means that the matrix is a singular matrix, and its determinant is equal to 0. This also means that the matrix is not invertible.

4. How can one prove that the determinant of an NxN matrix is 0 when all rows and columns sum to 0?

One can prove this by using the properties of determinants, such as the fact that the determinant of a triangular matrix is equal to the product of its diagonal elements. By using this property and the fact that the matrix has all rows and columns summing to 0, it can be shown that the determinant must be 0.

5. What is the significance of proving that the determinant of an NxN matrix is 0 when all rows and columns sum to 0?

This proof is significant because it shows that the matrix is not invertible and has no inverse matrix. This can be useful in various applications, such as solving systems of linear equations, where the determinant of the matrix determines whether a unique solution exists or not.

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