A note on the determinant 308 in Proskuryakov's linear algebra book

TL;DR

This paper identifies and corrects an error in a determinant formula from Proskuryakov's linear algebra book and uses it to reprove that the Fubini-Study metric on complex projective space is Einstein.

Contribution

It corrects a specific formula in a classical linear algebra text and demonstrates its application to a fundamental geometric property.

Findings

Corrected the determinant 308 formula in Proskuryakov's book

Reproved the Einstein property of the Fubini-Study metric using the corrected formula

Clarified the mathematical relationship between the determinant formula and geometric structures

Abstract

We put in evidence and correct a mistake in the formula for the determinant 308 in Proskuryakov's linear algebra book. We apply this formula to reprove the well-known fact that the Fubini-Study metric on the complex projective space is Einstein.