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Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

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A - [\begin{array}{cc} 0 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 0 \end{array}]

I + A = [\begin{array}{cc} 1 & a m p; 0 \\ 0 & a m p; 1 \end{array}] + [\begin{array}{cc} 0 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 0 \end{array}]

= [\begin{array}{cc} 1 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 1 \end{array}]

I - A = [\begin{array}{cc} \cos α,, & a m p; - \sin α,, \\ \sin α,, & a m p; \cos α,, \end{array}]

= [[\begin{array}{cc} 1 & a m p; 0 \\ 0 & a m p; 1 \end{array}] - [\begin{array}{cc} 0 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 0 \end{array}]] \times [\begin{array}{cc} \cos α,, & a m p; - \sin α,, \\ \sin α,, & a m p; \cos α,, \end{array}]

= [\begin{array}{cc} 1 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 1 \end{array}] \times [\begin{array}{cc} \cos α,, & a m p; - \sin α,, \\ \sin α,, & a m p; \cos α,, \end{array}]

= [\begin{array}{cc} \cos α, + \tan α / 2 \sin α,, & a m p; - \sin α 2 + \tan α / 2 \cos α,, \\ \tan α / 2 \cos α, + \sin α,, & a m p; \sin α, \tan α / 2 + \cos α,, \end{array}]

= [\begin{array}{cc} 1 - \tan \frac{α,}{2}, + \frac{\tan \frac{α,}{2},,}{1 + \tan \frac{2 α,}{2},,} \times 2 \tan \frac{α,}{2},, & a m p; - 2 \tan \frac{α,}{2}, + \frac{\tan \frac{α,}{2},,}{1 + \tan \frac{2 α,}{2},,} - \tan \frac{3 α,}{2},, \\ 2 \tan \frac{α,}{2}, - \frac{\tan \frac{α,}{2},,}{1 + \tan \frac{2 α,}{2},,} \tan \frac{3 α,}{2},, & a m p; 1 - \tan \frac{α,}{2}, + \frac{\tan \frac{α,}{2},,}{1 + \tan \frac{2 α,}{2},,} \times 2 \tan \frac{α,}{2},, \end{array}]

= [\begin{array}{cc} 1 & a m p; - \tan α / 2, \\ \tan α / 2, & a m p; 1 \end{array}]

= I + A

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