ThinkDifferent

(a) Let there be a binary(0, 1) matrix T[8*8] =

0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 1

Let me define:

1. a boolean product (x.y), for any two binary numbers x and y, as:
x.y = 1, if and only if x = y = 1

2. a boolean sum (x + y), for any two binary numbers x and y, as:
x + y = 0, if and only if x = y = 0

3. a Boolean-Matrix-Multiplication as the normal matrix multiplication, that uses boolean product and sum, instead of the usual (base-10) arithmetic product and sum respectively.

Find the minimum number m such that

T.T.T...(m times) = T.T.T...(m+k times)

for any integer k > 1, where T.T represents Boolean-Matrix-Multiplication of T with T.

(b) How (if at all) will the answer change, if T were a symmetric boolean matrix as:

0 1 0 0 0 0 0 0
1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0
0 0 1 0 1 0 0 0
0 0 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1
0 0 0 0 0 0 1 1

Hint: find

T.T.T...(m times) = T.T.T...(m+2k times)

for any integer k > 1.