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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

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.