Verified Solution • 2025–26 Session moderate

QJEEM24Ma1c94: JEE Main 2024 Mathematics - Class 12 JEEM24Ma1c94 Solution: Let be a square matrix such that . Then is equal to

Let A\mathrm{A} be a square matrix such that AAT=I\mathrm{AA}^{\mathrm{T}}=\mathrm{I} . Then 12A[(A+AT)2+(A−AT)2]\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right] is equal to

Multiple Choice Options

  • A A2+AT\mathrm{A}^2+\mathrm{A}^{\mathrm{T}}
  • B A3+I\mathrm{A}^3+\mathrm{I}
  • C A3+AT\mathrm{A}^3+\mathrm{A}^{\mathrm{T}}
  • D A2+I\mathrm{A}^2+\mathrm{I}

Step-by-Step Verified Solution

AAT=I=ATA\mathrm{AA}^{\mathrm{T}}=\mathrm{I}=\mathrm{A}^{\mathrm{T}} \mathrm{A}

On solving given expression, we get

amp;12 A[ A2+(AT)2+2 A AT+A2+(AT)2−2 A AT]amp;=A[A2+(AT)2]=A3+AT\begin{aligned}& \frac{1}{2} \mathrm{~A}\left[\mathrm{~A}^2+\left(\mathrm{A}^{\mathrm{T}}\right)^2+2 \mathrm{~A} \mathrm{~A}^{\mathrm{T}}+\mathrm{A}^2+\left(\mathrm{A}^{\mathrm{T}}\right)^2-2 \mathrm{~A} \mathrm{~A}^{\mathrm{T}}\right] \\& =\mathrm{A}\left[\mathrm{A}^2+\left(\mathrm{A}^{\mathrm{T}}\right)^2\right]=\mathrm{A}^3+\mathrm{A}^{\mathrm{T}}\end{aligned}

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