% #5
rref([0,3,-5;1,0,2;-4,-9,7])
% #9
A=[4,0,-7,-7;-6,1,11,9;7,-5,10,19;-1,2,3,-1]
rref(A)
% #42
x1 = rand(4,1)
b = A*x1
x2 = A\b
x1-x2

% "proof adjacency matrix":
M= [0,0,0,0,0,0,0,0,0,1,1,1;
    1,0,0,0,0,0,0,0,0,0,0,0;
    0,1,0,0,0,0,0,0,0,0,0,0;
    0,0,1,0,1,0,0,0,0,0,0,0;
    0,0,0,1,0,1,0,0,0,0,0,0;
    0,0,0,0,1,0,0,0,0,0,0,0;
    1,0,0,0,0,0,0,1,0,0,0,0;
    0,0,0,0,0,0,1,0,1,0,0,0;
    0,0,0,0,0,0,0,1,0,0,0,0;
    0,0,0,1,0,0,0,0,0,0,0,0;
    0,0,0,0,0,0,1,0,0,0,0,0;
    1,0,0,0,0,0,0,0,0,0,0,0;
    ]
M=transpose(M)

v=[1;0;0;0;0;0;0;0;0;0;0;0]
M*v

% Picked twelve as large enough...
M^12*v

v=[0;0;0;0;0;0;1;0;0;0;0;0]
M*v
M^12*v

M^12
% 10 is interesting! It's the last power with a zero in a component...
M^10

