A) 1
B) \[-1\]
C) 0
D) None of these
Correct Answer: C
Solution :
Given, \[B={{A}^{-1}}.{{A}^{T}}\] \[\Rightarrow \] \[{{B}^{T}}={{({{A}^{-1}}{{A}^{T}})}^{T}}=A.{{({{A}^{-1}})}^{T}}\] \[\Rightarrow \] \[B.{{B}^{T}}={{A}^{-1}}{{A}^{T}}=A.{{({{A}^{-1}})}^{T}}\] \[={{A}^{-1}}.({{A}^{T}}A){{({{A}^{-1}})}^{T}}\] \[={{A}^{-1}}(A.{{A}^{T}}){{({{A}^{-1}})}^{T}}\] \[=({{A}^{-1}}A){{({{A}^{-1}}A)}^{T}}\] \[=I{{(I)}^{T}}=I\] Now, \[\det (B-I)=\det (B-B{{B}^{T}})\] \[=\det (B).\det (I-{{B}^{T}})\] \[=\det (B).\det (I-B)\] \[=1.\det (B-I)\] \[=\det (B-I)=0\]You need to login to perform this action.
You will be redirected in
3 sec