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  4. If a square matrix A is such that AAT=I=ATA, then |A| is equal to:

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Q. If a square matrix A is such that $ A{{A}^{T}}=I={{A}^{T}}A, $ then $ |A| $ is equal to:

Bihar CECEBihar CECE 2006

Solution:

The Given, A is a square matrix and $ A{{A}^{T}}=I={{A}^{T}}A $
$ \Rightarrow $ $ |A{{A}^{T}}|=|I|=|{{A}^{T}}A| $
$ \Rightarrow $ $ |A||{{A}^{T}}|=1=|{{A}^{T}}||A| $
$ \Rightarrow $ $ |A{{|}^{2}}=1 $ $ (\because \,|{{A}^{T}}|=|A|) $
$ \Rightarrow $ $ |A|=\pm 1. $