Let a, b, c be any real numbers. Suppose that there are real...

Let $a, b, c$ be any real numbers. Suppose that there are real numbers $x, y, z$ not all zero such that $x=c y+b z, y$ $=a z+c x$ and $z=b x+a y$. Then $a^{2}+b^{2}+c^{2}+2 a b c$ is equal to
(A) 2
(B) $-1$
(C) 0
(D) 1

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Key Concepts

Symmetric Matrices

A symmetric matrix is one that is equal to its transpose, meaning its entries are mirrored along the main diagonal. Such matrices often have advantageous properties, including the guarantee of real eigenvalues and the potential for a simpler analysis of determinants. These properties are particularly useful when setting up and solving systems of equations derived from symmetric relationships.

Determinant of a Matrix

The determinant is a scalar value calculated from a square matrix that provides key insights into the matrix's properties. A zero determinant indicates that the matrix is singular, meaning it does not have an inverse and that the corresponding system of linear equations has either no unique solution or infinitely many solutions. In the context of homogeneous systems, setting the determinant to zero yields the necessary condition for nontrivial solutions.

Homogeneous Linear Systems

A homogeneous linear system is one in which all the constant terms are zero. Such systems always have the trivial solution where all variables equal zero. However, the existence of nontrivial solutions hinges on the structure of the coefficient matrix, and specifically on whether its determinant is zero. This concept is crucial for determining conditions under which a system of equations has solutions beyond the trivial one.

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