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)

nicekikah 2021-02-03 Answered
Which of the following statements are always true, with A and B are orthogonal matrices?
1)det(AT(A+B)BT)=det(AT+BT)
2)det(A1+BT)=det(AT+B1)
3)det(A+BT)=det(AT+B)
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Expert Answer

Nola Robson
Answered 2021-02-04 Author has 94 answers
Step 1
It is given that A and B are orthogonal matrices. The properties of orthogonal matrix are as follows:
AAT=ATA=I
AT=A1 Note that, AT(A+B)BTATABT+ATBBT
Thus, it is not true that 1)det(AT(A+B)BT)=det(AT+BT)
Hence, option 1 is incorrect.
Step 2
Note that AT=A1, thus |A1+BT|=|AT+B1|
Therefore, option 2 is correct.
Since A and B are orthogonal matrices, AAT and BBT
Thus, |A+BT||AT+B|
Hence, option 3 is also incorrect.
Thus, the correct option is 2.
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Jeffrey Jordon
Answered 2022-01-15 Author has 2087 answers

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