Question

If $straight A space equals space open square brackets table row 1 cell space space space space space tanx end cell row cell negative tanx end cell cell space space space 1 end cell end table close square brackets comma space space space space then space straight A apostrophe straight A to the power of negative 1 end exponent space equals space open square brackets table row cell cos space 2 straight x end cell cell space space space space space minus sin space 2 straight x end cell row cell sin space 2 straight x end cell cell space space space space space space space space cos space 2 straight x end cell end table close square brackets$

Solution

Here, $straight A space equals space open square brackets table row 1 cell space space space space tanx end cell row cell negative tanx end cell cell space space space 1 end cell end table close square brackets$
$therefore space space space open vertical bar straight A close vertical bar space equals space open vertical bar table row 1 cell space space space space tanx end cell row cell negative tanx end cell cell space space space 1 end cell end table close vertical bar space equals space 1 plus tan squared straight x space equals space sec squared straight x therefore space space space straight A to the power of negative 1 end exponent space exists. space$

Co-factors of the elements of first row are 1, tan x respectively.
Co-factors of the elements of second row are – tan x, 1 respectively.
$therefore space space space adj. space straight A space equals space open square brackets table row 1 cell space space space space tanx end cell row cell negative tanx end cell cell space space space 1 end cell end table close square brackets to the power of apostrophe space equals space open square brackets table row 1 cell negative tanx end cell row tanx cell space space space 1 end cell end table close square brackets therefore space space space space straight A to the power of negative 1 end exponent space equals space fraction numerator adj. space straight A over denominator open vertical bar straight A close vertical bar end fraction space equals space fraction numerator 1 over denominator sec squared straight x end fraction open square brackets table row 1 cell space space space space space minus tanx end cell row tanx cell space space space space space 1 end cell end table close square brackets Also comma space space space straight A apostrophe space equals space open square brackets table row 1 cell space space space space minus tanx end cell row tanx cell space space space space space space 1 end cell end table close square brackets therefore space space space straight A apostrophe straight A to the power of negative 1 end exponent space equals space open square brackets table row 1 cell space space space space minus tanx end cell row tanx cell space space space space 1 end cell end table close square brackets space fraction numerator 1 over denominator sec squared straight x end fraction open square brackets table row 1 cell space space space minus tanx end cell row tanx cell space space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator sec squared straight x end fraction open square brackets table row 1 cell space space space space minus tanx end cell row tanx cell space space space space space space space 1 end cell end table close square brackets space space open square brackets table row 1 cell space space space minus tanx end cell row tanx cell space space space space 1 end cell end table close square brackets space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator sec squared straight x end fraction open square brackets table row cell 1 minus tanx end cell cell space space space space space 1 minus 2 space tanx end cell row cell 2 space tanx end cell cell space space 1 minus tan squared straight x end cell end table close square brackets space space space space space space space space space space space space space space space space space equals space cos squared straight x open square brackets table row cell fraction numerator cos squared straight x minus sin squared straight x over denominator cos space straight x end fraction end cell cell negative fraction numerator 2 space sinx space over denominator cos space straight x end fraction end cell row cell fraction numerator 2 space sinx space over denominator cos space straight x end fraction end cell cell fraction numerator cos squared straight x minus sin squared straight x over denominator cosx end fraction end cell end table close square brackets space space space space space space space space space space space space space space space space space equals space open square brackets table row cell cos squared straight x minus sin squared straight x end cell cell space space space space space minus 2 space sin space straight x space cos space straight x end cell row cell 2 space sinx space cosx end cell cell space space space space space space cos squared straight x minus sin squared straight x end cell end table close square brackets therefore space space space space straight A apostrophe straight A to the power of negative 1 end exponent space equals space open square brackets table row cell cos space 2 straight x end cell cell space space space space space minus sin space 2 straight x end cell row cell sin space 2 straight x end cell cell space space space space space space space cos space 2 straight x end cell end table close square brackets$

For Daily Free Study Material Join wiredfaculty

Determinants

Some More Questions From Determinants Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction