Find the composite Transformation matrix?

Question

a) Find the composite matrix that performs the following
transformations ( in order )

rotate by 45 degrees along x-axis
scale by (2,2,3)
translate by (0,-2,3)

b) Apply this matrix to the positive unit cube with a vertex at (0,0,0), and give the new coordinates of all the cube vertices

PLEASE I NEED EXPLANATION AS MUCH AS COULD BE.
THANKS IN ADVANCE

Answer 1

Please check the reference below for construction of answer in a 4x4 matrix.

Answer 2

I assume that the two vertices on opposite ends of the body diagonal are at (0,0,0) and (1,1,1) to start.

1) Rotate by 45 degrees along x-axis.
Now they are at (0,0,0) and (1/√2, 1/√2, 1).

2) Scale by (2,2,3).
Now they are at
(2, 2, 3)*(0,0,0) = (0,0,0)
and
(2, 2, 3)*(1/√2, 1/√2, 1) = (√2, √2, 3).

3) Translate by (0,-2,3).
Now they are at
(0,0,0) + (0, -2, 3) = (0, -2, 3)
and
(√2, √2, 3) + (0, -2, 3) = (√2, -2+√2, 6)

The two vertices on opposite ends of the body diagonal are at (0, -2, 3) and (√2, -2+√2, 6).

The other six can be readily determined from these two.