{"id":20038,"date":"2024-04-15T05:47:32","date_gmt":"2024-04-15T05:47:32","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20038"},"modified":"2024-04-15T05:47:32","modified_gmt":"2024-04-15T05:47:32","slug":"the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real\/","title":{"rendered":"The eigen values of a skew-symmetric matrix are A. always zero B. always pure imaginary C. either zero or pure imaginary D. always real"},"content":{"rendered":"

[amp_mcq option1=”always zero” option2=”always pure imaginary” option3=”either zero or pure imaginary” option4=”always real” correct=”option3″]<\/p>\n

The correct answer is: C. either zero or pure imaginary.<\/p>\n

A skew-symmetric matrix is a square matrix $A$ such that $A^T = -A$. The eigenvalues of a skew-symmetric matrix are always either zero or pure imaginary. This can be shown by using the fact that the characteristic polynomial of a skew-symmetric matrix is of the form $p(x) = x^2 – tr(A)x + \\det(A)$. Since $tr(A) = 0$ for a skew-symmetric matrix, the characteristic polynomial has two distinct roots, which must be either both real or both complex. If the roots are real, then they must be both zero. If the roots are complex, then they must be a complex conjugate pair.<\/p>\n

Here is a brief explanation of each option:<\/p>\n