Let A be a square matrix such that A A^T=I. Then 1 2 A [ (A+A^T )… - Vidyadip

MCQ

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

A
$A^2+I$
B
$A^3+I$
C
$A^2+A^T$
✓
$A^3+A^T$

✓

Answer

Correct option: D.

$A^3+A^T$

d
$\mathrm{AA}^{\mathrm{T}}=\mathrm{I}=\mathrm{A}^{\mathrm{T}} \mathrm{A}$

On solving given expression, we get

$ \frac{1}{2} A\left[A^2+\left(A^T\right)^2+2 A A^T+A^2+\left(A^T\right)^2-2 A A^T\right] $

$ =A\left[A^2+\left(A^T\right)^2\right]=A^3+A^T$

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