Question

If M is a 3 × 3 matrix, where M^TM = I and det(M) = 1, then prove that det(M - I) = 0.

Answer 1

(M - I)^T = M^T - I = M^T - M^T M = M^T(I - M)

⇒ |(M - I)^T | = |M - I| = |M^T | |I - M| = |I - M|

⇒ |M - I| = 0

Alternate method

det(M - I) = det(M - I) det(M^T) = det(MM^T - M^T)

= det(I - M^T) = - det(M^T - I) = - det(M - I)^T = -det(M - I)

⇒ det(M - I) = 0