Verified Solution • 2025–26 Session moderate

QJEEM14Macf21: JEE Main 2014 Mathematics - Class 12 JEEM14Macf21 Solution: If is a non-singular matrix such that and then equals:

If AA is a 3×33 \times 3 non-singular matrix such that AA′=A′AAA'=A'A and
B=A−1A′,B = {A^{ - 1}}A', then BB′BB' equals:

Multiple Choice Options

  • A B−1{B^{ - 1}}
  • B (B−1)′\left( {{B^{ - 1}}} \right)'
  • C I+BI+B
  • D II

Step-by-Step Verified Solution

BB′=B(A−1A′)′=B(A′)′(A−1)′=BA(A−1)′BB' = B\left( {{A^{ - 1}}A'} \right)' = B\left( {A'} \right)'\left( {{A^{ - 1}}} \right)' = BA\left( {{A^{ - 1}}} \right)'

=(A−1A′)(A(A−1)′)= \left( {{A^{ - 1}}A'} \right)\left( {A\left( {{A^{ - 1}}} \right)'} \right)

=A−1A.A′.(A−1)′      = {A^{ - 1}}A.A'.\left( {{A^{ - 1}}} \right)'\,\,\,\,\,\, {\left\{ {} \right. as       \,\,\,\,\,\, AA′=A′AAA' = A'A  }\left. \, \right\}

=I(A−1A)′= I\left( {{A^{ - 1}}A} \right)'

=I.I=I2=I= I.I = {I^2} = I

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