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

If M is a `3xx3` matrix, where `det M = I and M M^(T) = I,`
where `I` is an identity matrix, prove that `det(M-I) = 0`

Text Solution

Verified by Experts

`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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