Question

Let M be an 4×4 matrix with real entries such that M^k=0, for some k≥1. Find the determinant value of (I+M), where, I be the 4 x 4 identity matrix.

(a) 72

(b) 1

(c) 4

(d) 36

I got this question in an online quiz.

I would like to ask this question from Permutation Groups topic in portion Groups of Discrete Mathematics

Answer 1

The correct answer is (b) 1

The explanation: By cayley hamilton theorem, M^4 = 0. So, characteristic equation should be λ*4=0 and after solving we get 0 for every eigen value. Eigen values of (I+M) = Individual Eigen value of 1+m. So all the eigen values of (I+M) are 1 and Det(I+A) = 1.