Linear algebra help?

Question

there was this really hard question on my exam. please help, thanks in advance

Problem: An n x n matrix A is said to be skew-symmetric if A^T = - A

a) By finding det A in two different ways, show that det A satisfies the the equation: det A = (-1)^n det A

b) Show that (you may use part a)

det A = 0 where A is the matrix below

0 1 3 -2 3

-1 0 6 -1 4

-3 -6 0 5 1

2 1 -5 0 0

-3 -4 -1 0 0

Answer 1

If A^t = −A, then:

(1) det(A^t) = det(−A)

But det(A^t) = det(A) and det(cA) = (c^n)det(A), if A is a n×n matrix, so (1) is equivalent to:

det(A) = (−1^n)det(A)

As for (b), you have an skew-symmetric 5×5 matrix, so det(A) = −det(A).