InterviewSolution
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InterviewSolution
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Let B is an invertible square matrix and B is the adjoint of matrix A such that `AB=B^(T)`. ThenA. A is an identity matrixB. B is symmetric matrixC. A is a skew-symmetric matrixD. B is skew symmetic matrix |
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Answer» 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` |
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