Null termination of char array

Consider following case:

#include<stdio.h>
int main()
{
    char A[5];
    scanf("%s",A);
    printf("%s",A);
}

My question is if char A[5] contains only two characters. Say "ab", then A[0]='a', A[1]='b' and A[2]='\0'. But if the input is say, "abcde" then where is '\0' in that case. Will A[5] contain '\0'? If yes, why? sizeof(A) will always return 5 as answer. Then when the array is full, is there an extra byte reserved for '\0' which sizeof() doesn't count?