Let B is an invertible square matrix and B is the adjoint of matrix A such that `AB=B^(T)`. Then

Options

A. A is an identity matrix
B. B is symmetric matrix
C. A is a skew-symmetric matrix
D. B is skew symmetic matrix

Answer

A

Explanation

Given that `AB=B^(T)` and `B=` adj. A
`implies A=B^(T)B^(-1)`
`implies |A|=|B^(T)|xx|B^(-1)|=1` ...(i)
Now, adj. `A=` adj `(B^(T) B^(-1))`
`="adj"(B^(-1))xx"adj"(B^(T))`
`=("adj B")^(-1)xx"adj"(B^(T))`
`implies ("adj. B")^(T)=("adj. B")xx("adj. A")=("adj. B")B=|B|I`
`implies ("adj. B")^(T)=|B|. I`
`implies ("adj. (adj. A)")^(T)=|"adj. A"|I`
`:. |A|^(n-2) A^(T)=|A|^(n-1)I`
`implies A^(T)=|A|I`
`implies A^(T)=I`
`implies A=I`
`:.` B=adj. A=adj. `I=I`