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Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

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A = (\begin{array}{cc} p & a m p; q \\ 0 & a m p; 1 \end{array})

then

A^{2} = (\begin{array}{cc} p & a m p; q \\ 0 & a m p; 1 \end{array}) (\begin{array}{cc} p & a m p; q \\ 0 & a m p; 1 \end{array})

= (\begin{array}{cc} p^{2} + 0 & a m p; p q + q \\ 0 + 0 & a m p; 0 + 1 \end{array})

= (\begin{array}{cc} p^{2} & a m p; p q + q \\ 0 & a m p; 1 \end{array})

A^{3} = A^{2} . A

= (\begin{array}{cc} p^{2} & a m p; p q + q \\ 0 & a m p; 1 \end{array}) (\begin{array}{cc} p & a m p; q \\ 0 & a m p; 1 \end{array})

= (\begin{array}{cc} p^{3} + 0 & a m p; p^{2} q + p q + q \\ 0 & a m p; 1 \end{array})

= (\begin{array}{cc} p^{3} & a m p; q (p^{2} + p + 1) \\ 0 & a m p; 1 \end{array})

A^{4} = A^{3} . A

= (\begin{array}{cc} p^{3} & a m p; q (1 + p + p^{2}) \\ 0 & a m p; 1 \end{array}) (\begin{array}{cc} p & a m p; q \\ 0 & a m p; 1 \end{array})

= (\begin{array}{cc} p^{4} & a m p; p^{3} q + q (1 + p + p^{2}) \\ 0 & a m p; 1 \end{array})

A^{4} = (\begin{array}{cc} p^{4} & a m p; q (1 + p + p^{2} + p^{3}) \\ 0 & a m p; 1 \end{array})

A^{8} = (\begin{array}{cc} p^{8} & a m p; q (1 + p + p^{2} + . . . . + p^{7}) \\ 0 & a m p; 1 \end{array})

By G.P

q (1 + p + p^{2} + . . . . + p^{7})

= q (\frac{p 8 - 1}{p - 1})

A^{8} = (\begin{array}{cc} p^{8} & a m p; q (\frac{p^{8} - 1}{p - 1}) \\ 0 & a m p; 1 \end{array})