If Ais symmetric and B is skew-symmetric matrix, then which of the following is/are CORRECT ?

Question

If Ais symmetric and B is skew-symmetric matrix, then which of the following is/are CORRECT ?

If Ais symmetric and B is skew-symmetric matrix, then which of the following is/are CORRECT ?
A. `ABA^(T)` is skew-symmetric matrix
B. `AB^(T)+BA^(T)` is symmetric matrix
C. `(A+B) (A-B)` is skew-symmetric
D. `(A+I) (B-I)` is symmetric

Monjur Alahi

6 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

`A^(T)=A` and `B^(T)=-B`
(1) `(ABA^(T))^(T)=AB^(T)A^(T)=-(ABA^(T))`
(2) `AB^(T)+BA^(T)=-AB+BA`
Now, `(BA-AB)^(T)=(BA)^(T)-(AB)^(T)`
`=A^(T)B^(T)-B^(T)A^(T)=-AB+BA=AB^(T)+BA^(T)`
(3) `((A+B)(A-B))^(T)=(A-B)^(T) (A+B)^(T)`
`=(A^(T)-B^(T)) (A^(T)+B^(T))=(A+B) (A-B)`
(4) `((A+I) (B-I))^(T)=(B^(T)-I) (A^(T)+I)`
`=-(-B+I) (A+I)=(B-I) (A+I)`