<s>[INST] Transformation Matrix for cube in 2D My task is to transform the cube from the left corner to the big cube in the middle:
What I did was:
First i scale the cube: 
$$
        \begin{pmatrix}
        4 & 0 & 0 \\
        0 & 4 & 0 \\
        0 & 0 & 1 \\
        \end{pmatrix}
$$
Next i rotate it $45^\circ$:
$$
        \begin{pmatrix}
        \cos(45^\circ) & -\sin(45^\circ) & 0 \\
        \sin(45^\circ) & \cos(45^\circ) & 0 \\
        0 & 0 & 1 \\
        \end{pmatrix}
$$
And last i translate the cube:
$$
        \begin{pmatrix}
        1 & 0 & 10 \\
        0 & 1 & 7 \\
        0 & 0 & 1 \\
        \end{pmatrix}
$$
When i  combine this matrices i get my transformation matrix:
$$
        \begin{pmatrix}
        4\cos(45^\circ) & -4\sin(45^\circ) & 40\cos(45^\circ) - 28\sin(45^\circ) \\
        4\sin(45^\circ) & 4\cos(45^\circ) & 40\cos(45^\circ) + 28\sin(45^\circ) \\
        0 & 0 & 1 \\
        \end{pmatrix}
$$
But then i tried to apply this matrix to the point $(0,1,1)$ and i got an wrong result:
$$(5.6, 50.9116 ,1)$$
As you can see it should be around:
$$(7.1, 9.9, 1)$$
What did I wrong? Thanks!
 [/INST] There is a mistake in your multiplication.
$$
        T_{total}=\begin{pmatrix}
        4\cos(45^\circ) & -4\sin(45^\circ) & 40\cos(45^\circ) - 28\sin(45^\circ) \\
        4\sin(45^\circ) & 4\cos(45^\circ) & \color{red}{28\cos(45^\circ) + 40\sin(45^\circ)} \\
        0 & 0 & 1 \\
        \end{pmatrix}
$$
But, since $sin (45^\circ)$ and $cos (45^\circ)$ have the same value, the problem is not from this point. 
The problem is related to the order of the Transforms. In matrix multiplication, you should care about matrix order.
$T1$ is scale transform
$T2$ is rotation transform
$T3$ is translation transform
you computed:
$$X_{new}=(T1 \times T2 \times T3) \times X_{old}$$
while, you need to scale first then rotation then translation:
$$X_{new}=(T3 \times T2 \times T1) \times X_{old}$$
$$T3 \times T2 \times T1=\begin{pmatrix}
        4\cos(45^\circ) & -4\sin(45^\circ) & 10 \\
        4\sin(45^\circ) & 4\cos(45^\circ) & 7 \\
        0 & 0 & 1 \\
        \end{pmatrix}$$
$$T1\times T2 \times T3 \times \begin{pmatrix}
        0 \\
        1 \\
        1 \\
        \end{pmatrix}=\begin{pmatrix}
        5.66 \\
        50.91 \\
        1 \\
        \end{pmatrix}$$
While
$$T3\times T2 \times T1 \times \begin{pmatrix}
        0 \\
        1 \\
        1 \\
        \end{pmatrix}=\begin{pmatrix}
        7.17 \\
        9.83 \\
        1 \\
        \end{pmatrix}$$
</s>