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R.P.
R.P.
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Prove that there is a diagonal matrix $D$ with entries equal to $\pm 1$ diagonal entries such that $\det(A+D) \neq 0$

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edit approved Jun 3, 2017 at 14:14
Rodrigo de Azevedo
edit approved Jun 3, 2017 at 14:14
Rodrigo de Azevedo
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Prove that there is a diagonal matrix $D$ with entries $\pm 1$ with $detdiagonal entries such that $\det(A+D) \neq 0$

Lately I recently saw athe following problem ofon an entrance exam as following:

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $det(A+D)\neq 0$.

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $\det(A+D) \neq 0$.

I totally have no idea how to deal with the determinant of a sum of matrices. I think it can be proved by contradiction. But I don't see what would lead to if $det(A+D)=0$$\det(A+D)=0$ for all diagonal matrix $D$ with diagonal entries $\pm1$$\pm 1$.

  Hope someone could help me with this one, thanks. Thanks!

Prove that there is a diagonal $D$ with entries $\pm 1$ with $det(A+D) \neq 0$

Lately I saw a problem of entrance exam as following:

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $det(A+D)\neq 0$.

I totally have no idea how to deal with determinant of sum of matrices. I think it can be proved by contradiction. But I don't see what would lead to if $det(A+D)=0$ for all diagonal matrix $D$ with diagonal entries $\pm1$.

  Hope someone could help me with this one, thanks!

Prove that there is a diagonal matrix $D$ with $\pm 1$ diagonal entries such that $\det(A+D) \neq 0$

I recently saw the following problem on an entrance exam:

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $\det(A+D) \neq 0$.

I have no idea how to deal with the determinant of a sum of matrices. I think it can be proved by contradiction. But I don't see what would lead to if $\det(A+D)=0$ for all diagonal matrix $D$ with diagonal entries $\pm 1$. Hope someone could help me with this one. Thanks!

Post Closed as "Not suitable for this site" by Anthony Quas, Ryan Budney, Andreas Thom, Alexandre Eremenko, Chris Godsil
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user50190
user50190
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Prove that there is a diagonal $D$ with entries $\pm 1$ with $det(A+D) \neq 0$

Lately I saw a problem of entrance exam as following:

Let $A$ be a square matrix. Prove that there is a diagonal matrix $D$ whose diagonal entries are either $+1$ or $-1$ such that $det(A+D)\neq 0$.

I totally have no idea how to deal with determinant of sum of matrices. I think it can be proved by contradiction. But I don't see what would lead to if $det(A+D)=0$ for all diagonal matrix $D$ with diagonal entries $\pm1$.

Hope someone could help me with this one, thanks!