Solving Matrix Multiplication: Finding E(α) E(β)

We are given a matrix \({\rm{E}}\left( {\rm{\theta }} \right)\) defined as:

We need to find the product of \({\rm{E}}\left( {\rm{\alpha }} \right)\) and \({\rm{E}}\left( {\rm{\beta }} \right)\). Based on the definition of \({\rm{E}}\left( {\rm{\theta }} \right)\), we can write \({\rm{E}}\left( {\rm{\alpha }} \right)\) and \({\rm{E}}\left( {\rm{\beta }} \right)\) by replacing \({\rm{\theta }}\) with \({\rm{\alpha }}\) and \({\rm{\beta }}\) respectively.

Performing Matrix Multiplication E(α) E(β)

Now, let's multiply \({\rm{E}}\left( {\rm{\alpha }} \right)\) by \({\rm{E}}\left( {\rm{\beta }} \right)\). To multiply two \(2 \times 2\) matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix.

Let the resulting matrix be \({\rm{R}} = {\rm{E}}\left( {\rm{\alpha }} \right){\rm{E}}\left( {\rm{\beta }} \right) = \left[ {\begin{array}{*{20}{c}} {{{\rm{r}}_{11}}}&{{{\rm{r}}_{12}}}\\ {{{\rm{r}}_{21}}}&{{{\rm{r}}_{22}}} \end{array}} \right]\). We calculate each element:

So, the resulting matrix is:

Applying Trigonometric Identities

We can simplify the elements of this matrix using the angle addition formulas from trigonometry:

• Cosine addition formula: \(\cos({\rm{A}} + {\rm{B}}) = \cos {\rm{A}}\cos {\rm{B}} - \sin {\rm{A}}\sin {\rm{B}}\)
• Sine addition formula: \(\sin({\rm{A}} + {\rm{B}}) = \sin {\rm{A}}\cos {\rm{B}} + \cos {\rm{A}}\sin {\rm{B}}\)

Using these identities with \({\rm{A}} = {\rm{\alpha}}\) and \({\rm{B}} = {\rm{\beta}}\), we can rewrite the matrix elements:

• \({\rm{r}}_{11} = \cos {\rm{\alpha}}\cos {\rm{\beta}} - \sin {\rm{\alpha}}\sin {\rm{\beta}} = \cos({\rm{\alpha}} + {\rm{\beta}})\)
• \({\rm{r}}_{12} = \cos {\rm{\alpha}}\sin {\rm{\beta}} + \sin {\rm{\alpha}}\cos {\rm{\beta}} = \sin({\rm{\alpha}} + {\rm{\beta}})\)
• \({\rm{r}}_{21} = -(\sin {\rm{\alpha}}\cos {\rm{\beta}} + \cos {\rm{\alpha}}\sin {\rm{\beta}}) = -\sin({\rm{\alpha}} + {\rm{\beta}})\)
• \({\rm{r}}_{22} = \cos {\rm{\alpha}}\cos {\rm{\beta}} - \sin {\rm{\alpha}}\sin {\rm{\beta}} = \cos({\rm{\alpha}} + {\rm{\beta}})\)

Substituting these back into the matrix, we get:

Comparing with E(θ) Definition

Let's compare this resulting matrix with the original definition of \({\rm{E}}\left( {\rm{\theta }} \right)\). If we replace \({\rm{\theta }}\) with \({\rm{\alpha}} + {\rm{\beta}}\) in the expression for \({\rm{E}}\left( {\rm{\theta }} \right)\), we get:

We can see that the result of \({\rm{E}}\left( {\rm{\alpha }} \right){\rm{E}}\left( {\rm{\beta }} \right)\) is exactly the same as \({\rm{E}}\left( {\rm{\alpha }} + {\rm{\beta }} \right)\).

Conclusion

Therefore, \({\rm{E}}\left( {\rm{\alpha }} \right){\rm{E}}\left( {\rm{\beta }} \right)\) is equal to \({\rm{E}}\left( {\rm{\alpha }} + {\rm{\beta }} \right)\).

Revision Table: Matrix and Trigonometry Review

Additional Information on Rotation Matrices

The matrix \({\rm{E}}\left( {\rm{\theta }} \right) = \left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{\sin {\rm{\theta }}}\\ { - \sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\) is a standard representation of a 2D rotation matrix. Specifically, it represents a clockwise rotation by angle \({\rm{\theta }}\) about the origin. A counterclockwise rotation matrix is given by \(\left[ {\begin{array}{*{20}{c}} {\cos {\rm{\theta }}}&{ - \sin {\rm{\theta }}}\\ {\sin {\rm{\theta \;}}}&{\cos {\rm{\theta }}} \end{array}} \right]\). The given matrix has \(+\sin {\rm{\theta }}\) in the top right and \(-\sin {\rm{\theta }}\) in the bottom left, which corresponds to a clockwise rotation by \({\rm{\theta }}\). When you apply a rotation by \({\rm{\alpha }}\) followed by a rotation by \({\rm{\beta }}\), the net effect is a rotation by \({\rm{\alpha}} + {\rm{\beta}}\). This geometric property explains why multiplying the matrices \({\rm{E}}\left( {\rm{\alpha }} \right)\) and \({\rm{E}}\left( {\rm{\beta }} \right)\) results in \({\rm{E}}\left( {\rm{\alpha }} + {\rm{\beta }} \right)\). This result is consistent regardless of whether the matrix represents clockwise or counterclockwise rotation, as multiplying rotations corresponds to adding angles.