The rank of the matrix given below is: $$$\left[ {\matrix{ 1 & 4 & 8 & 7 \cr 0 & 0 & 3 & 0 \cr 4 & 2 & 3 & 1 \cr 3 & {12} & {24} & {2} \cr } } \right]$$$

Consider the following set a equations
x + 2y = 5
4x + 8y = 12
3x + 6y + 3z = 15 This set

Let (A, *) be a semigroup. Furthermore, for every a and b in A, if $$a\, \ne \,b$$, then $$a\,*\,b \ne \,\,b\,*\,a$$.

(a) Show that for every a in A
a * a = a
(b) Show that for every a, b in A
a * b * a = a
(c) Show that for every a, b, c in A
a * b * c = a * c

A die is rolled three times. The probability that exactly one odd number turns up among the three outcomes is