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If M is a 3xx3 matrix, where det M=1a n ...

If `M` is a `3xx3` matrix, where det `M=1a n dM M^T=1,w h e r eI` is an identity matrix, prove theat det `(M-I)=0.`

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We are given that `M M^(T)=I`, where M is a square matrix of order 3 and det. `M=1`.
Now, det. `(M-I)`
= det. `(M-M M^(T))" "[ :'" Given "M M^(T)=I]`
= det. `[M(I-M^(T))]`
`= ("det. M") ("det. "(I-M^(T)))`
`=-("det. M") ("det. "(M^(t)-I))" "[ :' M^(T)-I" has order 3"]`
`=-"det. "(M^(T)-I)" "[ :'" det. "M=1]`
`=-"det. "(M-I)" "[ :' "det. "(M^(T)-I)=" det. "(M-I)]`
`implies 2" det. "(M-I)=0`
`implies" det. "(M-I)=0`
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