If the elements of a matrix `A` are real positive and distinct such that `det(A+A^(T))^(T)=0` then

Options

A. `detA gt 0`
B. `det A ge 0`
C. `det (A-A^(T)) gt 0`
D. `det (A.A^(T)) gt 0`

Answer

A, C, D

Explanation

`(a,c,d)` `A=({:(a,b),(c,d):})` `a ne b ne c ne d gt 0`
`A+A^(T)=({:(2a,b+c),(b+c,2d):})`
`|A+A^(T)|=4ad-(b+c)^(2)=0impliesb+c=2sqrt(ad)`
`impliesb+c=2sqrt(ad) gt2sqrt(bc)` `(A.M gt G.M)`
`impliesad gt bc`
`impliesad-bc gt 0` (as `a ne b ne c ne d gt 0`)
`impliesdetA gt0`
`|A-A^(T)|=|{:((0,b-c),(c-b,0)):}|=0+(b-c)^(2) gt 0`
`|A A^(T)|=|A||A^(T)|=|A|^(2)=(det A)^(2) gt 0`