If A is symmetric and B skew- symmetric matrix and `A + B ` is non-singular and `C= (A+B) ^(-1) (A-B)` `C^(T) AC` equals to

Question

If A is symmetric and B skew- symmetric matrix and `A + B ` is non-singular and `C= (A+B) ^(-1) (A-B)` `C^(T) AC` equals to

If A is symmetric and B skew- symmetric matrix and `A + B ` is
non-singular and `C= (A+B) ^(-1) (A-B)`
`C^(T) AC` equals to
A. `A + B`
B. `A-B`
C. A
D. B

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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

Salim Ahmed Kawsar

Correct Answer - C
Given, `A^(T) = A, B^(T) = -B, det (A+ B) ne 0`
and `C = (A + B)^(-1) (A-B)`
`rArr (A + B) C = A - B` ...(i)
Also, ` (A + B) ^(T)= A - B` ...(ii)
and ` (A - B) ^(T)= A + B` ...(iii)
`C^(T) AC = C^(T)((A+B +A-B)/2) C`
`= 1/2 C^(T) (A + B) C+ 1/2 C^(T) (A-B) C`
`= 1/2 (A + B) + 1/2 (A-B) `
`=A`

4 views Answered 2023-02-05 15:29