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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.`

Text Solution

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`because MM^(T) = I" " ...(i)`
Let `B= M-I" "...(ii)`
`therefore B^(T) = M^(T) - I^(T) = M^(T) - M^(T)M" "` [from Eq. (i)]
` =M^(T) (I-M)= - M^(T)B ` [from Eq. (ii)]
Now, `det(B^(T)) = det(-M^(T)B)`
`=(-1)^(3)det(M^(T)) det(B) =-det(M^(T))det(B)`
`rArr det(B) =- det(M) det(B)=-det(B)`
`therefore det(B) = 0 `
`rArr det(M-I)=0`
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