Question

$If space straight A space equals space open parentheses table row cell cos space straight alpha end cell cell sin space straight alpha end cell row cell negative sin space straight alpha end cell cell cos space straight alpha end cell end table close parentheses comma space find space straight alpha space satisfying space 0 space less than space straight alpha space less than space straight pi over 2 space when space straight A space plus space straight A to the power of straight T space space equals space square root of 2 straight I subscript 2 semicolon space Where space straight A to the power of straight T space is space transpose space of space straight A.$

Solution

straight A space equals space open square brackets table row cell cos space straight alpha end cell cell sin space straight alpha end cell row cell negative sin space straight alpha end cell cell cos space straight alpha end cell end table close square brackets space space 0 space less than space straight alpha space less than space straight pi over 2 straight A space plus space straight A to the power of straight T space equals space square root of 2 straight I subscript 2 space open square brackets table row cell cos space straight alpha end cell cell sin space straight alpha end cell row cell negative sin space straight alpha end cell cell cos space straight alpha end cell end table close square brackets space plus space open square brackets table row cell cos space straight alpha end cell cell negative sin space straight alpha end cell row cell sin space straight alpha end cell cell cos space straight alpha end cell end table close square brackets space space equals space square root of 2 space end root open square brackets table row 1 0 row 0 1 end table close square brackets open square brackets table row cell 2 cos space straight alpha end cell 0 row 2 cell 2 cos space straight alpha end cell end table close square brackets space space equals space open square brackets table row cell square root of 2 end cell 0 row 0 cell square root of 2 end cell end table close square brackets 2 space cos space straight alpha space equals space square root of 2 cos space straight alpha space space equals space fraction numerator square root of 2 over denominator 2 end fraction space equals space fraction numerator 1 over denominator square root of 2 end fraction straight alpha space equals space straight pi over 4

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