# Which of the following statements are always true, with A and B are orthogonal matrices? 1)det(A^T(A+B)B^T)=det(A^T+B^T) 2)det(A^{-1}+B^T)=det(A^T+B^{-1}) 3)det(A+B^T)=det(A^T+B)

Which of the following statements are always true, with A and B are orthogonal matrices?
$1\right)det\left({A}^{T}\left(A+B\right){B}^{T}\right)=det\left({A}^{T}+{B}^{T}\right)$
$2\right)det\left({A}^{-1}+{B}^{T}\right)=det\left({A}^{T}+{B}^{-1}\right)$
$3\right)det\left(A+{B}^{T}\right)=det\left({A}^{T}+B\right)$
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Step 1
It is given that A and B are orthogonal matrices. The properties of orthogonal matrix are as follows:
$A{A}^{T}={A}^{T}A=I$
${A}^{T}={A}^{-1}$ Note that, ${A}^{T}\left(A+B\right){B}^{T}\ne {A}^{T}A{B}^{T}+{A}^{T}B{B}^{T}$
Thus, it is not true that $1\right)det\left({A}^{T}\left(A+B\right){B}^{T}\right)=det\left({A}^{T}+{B}^{T}\right)$
Hence, option 1 is incorrect.
Step 2
Note that
Therefore, option 2 is correct.
Since A and B are orthogonal matrices,
Thus, $|A+{B}^{T}|\ne |{A}^{T}+B|$
Hence, option 3 is also incorrect.
Thus, the correct option is 2.
Jeffrey Jordon