The value of the determinant

step 1 In problem 55 we want to find the value of the determinant 0 x y z minus x 0 c b minus y minus c

The value of the determinant $\left|\begin{array}{cccc}0 & x & y & z \\ -x & 0 & c & b \\ -y & -c & 0 & a \\ -z & -b & -a & 0\end{array}\right|$ is
(A) $(a x+b y+c z)^{2}$
(B) $(a x-b y+c z)^{2}$
(C) $(a x+b y-c z)^{2}$
(D) None of these

Key Concepts

Skew-Symmetric Matrix

A skew-symmetric matrix is one that satisfies the condition A^T = -A, meaning that the element in the i-th row and j-th column is the negative of the element in the j-th row and i-th column. This property implies that all the diagonal elements of a skew-symmetric matrix must be zero and that such matrices have special symmetry properties that affect their determinants and eigenvalues.

Pfaffian

The Pfaffian is a scalar associated with an even-dimensional skew-symmetric matrix, and it has the special property that the square of the Pfaffian equals the determinant of the matrix. This function simplifies the computation of determinants for skew-symmetric matrices and provides insights into their structural properties, making it an essential concept in linear algebra.

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