Let a be a matrix of order n, n be an odd number, satisfy AA ^ t = e, / A / = 1, find / a-e/

A-E = a-aa ^ t = a (e-A ^ t) = a (e-A) ^ t, take determinants on both sides, get
|A－E|＝|A|×|(E－A)^T|＝|E－A|＝(-1)^n×|A－E|＝－|A－E|
So | A-E | = 0

Let n-dimensional row vector, matrix A = e + 2AA ^ t, B = e - AA ^ t, where e is the unit matrix of order n, then a B=

AB = (E+2aa^T)(E-aa^T)
= E + 2aa^T - aa^T - 2aa^Taa^T
= E + aa^T - 2 (a^Ta) aa^T
= E + (1 - 2a^Ta) aa^T.

Is there a formula for the nth power of the following matrix
0 0 a
0 b 0
c 0 0

Let a = 0.0A
0 b 0
c 0 0
It is easy to find that a ^ 2 = AA = AC 0
0 b^2 0
0 0 ac
So a ^ (2n) = (AC) ^ n 0
0 b^(2n) 0
0 0 (ac)^n (n=1,2,.)
A^(2n+1)=A^(2n)*A=(ac)^(n) 0 0 0 0 a
0 B ^ (2n) 0 times 0 b 0
0 0 (ac)^(n) c 0 0
=0 0 a(ac)^(n)
0 b^(2n+1) 0
c(ac)^(n) 0 0 (n=1,2 ,.)

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Let a be a matrix of order n, n be an odd number, satisfy AA ^ t = e, / A / = 1, find / a-e/