recategorized by
682 views
1 votes
1 votes

Consider the polynomial $p(x)=x^3-x^2+x-1$. How many symmetric matrices with integer entries are there whose characteristic polynomial is $p$? (Recall that the characteristic polynomial of a square matrix $A$ in the variable $x$ is defined to be the determinant of the matrix $(A-x I)$ where $I$ is the identity matrix.)

  1. $0$
  2. $1$
  3. $2$
  4. $4$
  5. Infinitely many
recategorized by

1 Answer

0 votes
0 votes
All eigenvalues of a real symmetric matrix are real.

The characteristic polynomial of a square matrix has the eigenvalues as roots.

Roots of the given polynomial are 1, i, -i (non real).

Hence NO symmetric matrice with integer entries is there having given characteristic polynomial.
Answer:

Related questions

0 votes
0 votes
1 answer
2
admin asked Apr 1, 2020
581 views
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The valu...
1 votes
1 votes
2 answers
3
admin asked Apr 1, 2020
1,423 views
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiple by $2,$ its determinant value becomes $40.$ The value ...
1 votes
1 votes
0 answers
4
admin asked Sep 1, 2022
341 views
Let $\mathbb{R}$ denote the set of real numbers. Let $d \geq 4$ and $\alpha \in \mathbb{R}$. Let$$ S=\left\{\left(a_0, a_1, \ldots, a_d\right) \in \mathbb{R}^{d+1}: \sum_...