Matrix is a Set of similar type of values. In mathematics, matrix is used in various applications like engineering mathematics, trigonometry, geometry, differential equation, etc. We can convert an object in 3D matrix into translate matrix or in the rotation matrix. Rotation matrix is used in the pair of matrix. Translate matrix (T) and rotation matrix (R) when combines then they make transform matrix (Tr). |

**A Rotation Matrix is a three dimensional matrix in space. It is a displacement of a rigid body along with a fixed Point**.

**A body rotates about a fixed point that is known as a rotation**. This rotation can be expressed in the form of matrix also; this is known as rotation matrix. We are here to discuss rotation matrix proof.

We can prove this with the help of Euler theorem which states that it is possible to find the Diameter of a Sphere, when it rotates around its center and direction of the movement is same as initial direction.

Let us see rotation matrix derivation:

To prove this let us assume a matrix of 3 * 3 we call it 'R', it moves from the coordinate 'x' to 'X', which is Rx =X. For 'n' number of rotations we can calculate the number of rotation as Rn = n.

According to the rotation matrix property the multiplication of a rotation matrix and its transpose makes a identity matrix.

R . R

^{T}= I,

If there rotation matrix is of order 3 * 3 then its identity matrix is also of 3 * 3 order. When we find the determinant from this relation we get the determinant as ±1.

Here determinant +1 denotes perfect rotation and negative determinant denotes improper rotation. Now we know that every rotation matrix has at least one vector as 'n' which means Rn = n. This requires (R – I) n = 0. If n = 1 then the det (R - I) = 0.

These two Relations are used to prove Euler theorem; that is det (-R) = - det(R) and det(R - 1) = 1.

With help of these rules we get det (R - ƛ I) = 0, for ƛ = 1.

For 'n' number of rotations we get (R – I) n = 0 which is equals to Rn = n.

This is the proof of Euler’s theorem for derivation of rotation matrix.