Summary: LINEAR ALGEBRA (MATH 317H)
Assignment 14 - Determinants
(1) A square (n × n) matrix A is called skew-symmetric (or antisymmetric) if
= -A. Prove that if A is skew-symmetric and n is odd then det A = 0.
Is this also true for even n ?
(2) A square matrix A is called nilpotent if there is an integer k 1 for which
= 0. Show that if A is nilpotent then det A = 0.
(3) Prove that if matrices A and B are similar then det A = det B.
(4) A square matrix Q is called orthogonal if QT
Q = I. Prove that if Q is an
orthogonal matrix then det Q = 1 or det Q = -1.
(5) Use column or row operations to compute the following determinant
1 0 -2 3
-3 1 1 2
0 4 -1 1
2 3 0 1