TIFR CSE 2022 | Part A | Question: 4

Question

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.)

• A. $0$
• B. $1$
• C. $2$
• D. $4$
• E. Infinitely many

Answer 1

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.

Comments

What if the roots are real??