Question

$Let space straight A equals space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space space and space straight I comma space the space identity space matrix space of space order space 2. show space that space straight I plus straight A equals space left parenthesis straight I plus straight A right parenthesis space open square brackets table row cell cos space straight alpha end cell cell space space minus sin space straight alpha end cell row cell sin space straight alpha end cell cell space space space space cos space straight alpha end cell end table close square brackets$

Solution

Here space straight A equals space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space comma space straight I space equals space open square brackets table row 1 cell space space 0 end cell row 0 cell space space 1 end cell end table close square brackets straight L. straight H. straight S equals space straight I plus straight A space space space open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space 1 end cell end table close square brackets space space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space equals space space open square brackets table row 1 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 1 end cell end table close square brackets straight R. straight H. straight S equals space left parenthesis straight I minus straight A right parenthesis space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space open parentheses space open square brackets table row 1 cell space space space 0 end cell row 0 cell space space space 1 end cell end table close square brackets minus space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space close parentheses space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space space space space open square brackets table row 0 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 0 end cell end table close square brackets space space open square brackets table row cell cos space straight alpha end cell cell space space space space minus sin space straight a end cell row cell sin space straight alpha end cell cell space space space space cos space straight a end cell end table close square brackets space space space space space space space space space space space space equals space open square brackets table row cell cos space straight alpha space plus sin space straight alpha space tan begin inline style straight a over 2 end style space space space end cell cell negative sin space straight alpha space plus space cos space straight alpha space space tan begin inline style straight a over 2 end style end cell row cell negative cosα space tan begin inline style straight a over 2 space plus space sin space straight alpha end style end cell cell sin space straight alpha space tan straight alpha over 2 plus space cos space straight alpha end cell end table close square brackets equals space open square brackets table row cell 1 minus 2 space sin squared space begin inline style straight alpha over 2 end style space plus space 2 space sin squared space space begin inline style straight alpha over 2 end style space end cell cell space space space space space space minus tan space begin inline style straight alpha over 2 space open parentheses table row cell 2 cos squared space space space minus straight alpha over 2 end cell end table cosα space close parentheses end style end cell row cell tan space begin inline style straight alpha over 2 end style open parentheses table row cell 2 cos squared space begin inline style straight alpha over 2 end style space space minus end cell end table cosα space close parentheses end cell cell space 1 minus 2 space sin squared space begin inline style straight alpha over 2 end style space plus space 2 space sin squared space space begin inline style straight alpha over 2 end style end cell end table space space space space close square brackets equals space space open square brackets table row 1 cell space space space space minus tan space begin inline style straight a over 2 end style end cell row cell tan begin inline style straight alpha over 2 end style end cell cell space space 1 end cell end table close square brackets therefore straight L. straight H. straight S equals straight R. straight H. straight S

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Matrices

Some More Questions From Matrices Chapter

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

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If a matrix A has 12 elements, what arc the possible orders it can have 7 What if it has 7 elements ?

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