Question

Suppose, M is a lower triangular matrix with all diagonal entries zero. The resultant matrix of M+I will be ___________

(a) idempotent

(b) singular

(c) nilpotent

(d) inverse

I have been asked this question in final exam.

My doubt is from Permutation Groups topic in section Groups of Discrete Mathematics

Answer 1

Correct answer is (b) singular

Best explanation: Since, M is a lower triangular matrix with diagonal elements zero, then we add I and it will result in a lower triangular matrix with all diagonal entries 1. Thus, all eigenvalues M+I are non zero (eigenvalues of triangular matrix is the diagonal elements). So, determinant will never be zero. Hence, the matrix can have inverse property.