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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

Matrix Real - Rotate around X Axis

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

clear all;

% vertices of the cube

vert = [-0.5 -0.5 -0.5;  ...

        -0.5 0.5 -0.5;  ...

         0.5 0.5 -0.5;  ...

         0.5 -0.5 -0.5; ...

        -0.5 -0.5 0.5; ...

         -0.5 0.5 0.5;  ...

          0.5 0.5 0.5; ...

          0.5 -0.5 0.5];

fac = [1 2 3 4; ...

    2 6 7 3; ...

    4 3 7 8; ...

    1 5 8 4; ...

    1 2 6 5; ...

    5 6 7 8];

hFig = figure(1,'Position',[300 300 600 500]);

% transformed object

% This code add a vector to each vertices of the cube and store the result into TxVertices

phi = pi/6;

clear all;

% vertices of the cube

vert = [-0.5 -0.5 -0.5;  ...

        -0.5 0.5 -0.5;  ...

         0.5 0.5 -0.5;  ...

         0.5 -0.5 -0.5; ...

        -0.5 -0.5 0.5; ...

         -0.5 0.5 0.5;  ...

          0.5 0.5 0.5; ...

          0.5 -0.5 0.5];

fac = [1 2 3 4; ...

    2 6 7 3; ...

    4 3 7 8; ...

    1 5 8 4; ...

    1 2 6 5; ...

    5 6 7 8];

hFig = figure(1,'Position',[300 300 600 500]);

% transformed object

% This code add a vector to each vertices of the cube and store the result into TxVertices

n = 20;

phi = n * pi/10;

TxMatrix = [1     0          0 ; ...

            0    cos(phi)   -sin(phi); ...

            0    sin(phi)   cos(phi)];

TxVertices = TxMatrix * vert';

TxVertices = TxVertices';

% transformed object

subplot(2,2,1);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(30,30);

title('transformed');

xlabel('x');ylabel('y');zlabel('z');

tStr = sprintf("rotating angle = (%d pi)/10",n);

title(tStr);

% view along y-axis (x-z plane)

subplot(2,2,2);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(0,0);

title('x-z plane');

xlabel('x');ylabel('y');zlabel('z');

% view along x-axis (y-z plane)

subplot(2,2,3);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(90,0);

title('y-z plane');

xlabel('x');ylabel('y');zlabel('z');

% view along z-axis (x-y plane)

subplot(2,2,4);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(0,90);

title('x-y plane');

xlabel('x');ylabel('y');zlabel('z');

Sx = 1.0;

Sy = 1.0;

Sz = 1.0;

Sh12 = 0.0;

Sh13 = 0.0;

Sh21 = 0.0;

Sh23 = 0.0;

Sh31 = 0.0;

Sh32 = 0.0;

TxMatrix = [Sx     Sh12      Sh13 ; ...

            Sh21    Sy       Sh23; ...

            Sh31    Sh32       Sz];

TxVertices = TxMatrix * vert';

TxVertices = TxVertices';

% transformed object

subplot(2,2,1);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(30,30);

title('transformed');

xlabel('x');ylabel('y');zlabel('z');

tStr = sprintf("Sx = %0.1f, Sy = %0.1f, Sz = %0.1f",Sx,Sy,Sz);

title(tStr);

% view along y-axis (x-z plane)

subplot(2,2,2);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(0,0);

title('x-z plane');

xlabel('x');ylabel('y');zlabel('z');

% view along x-axis (y-z plane)

subplot(2,2,3);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(90,0);

title('y-z plane');

xlabel('x');ylabel('y');zlabel('z');

% view along z-axis (x-y plane)

subplot(2,2,4);

patch('Faces',fac,'Vertices',TxVertices,'FaceColor','b');  

axis([-2 2 -2 2 -2 2]);

grid();

view(0,90);

title('x-z plane');

xlabel('x');ylabel('y');zlabel('z');