Concept:
Nilpotent Matrix: Any square matrix of order n is said to be nilpotent matrix if there exist least positive integer m such that Am = O, where O is the null matrix of order n.
The determinant of the sum of the nilpotent matrix with the identity matrix of the same order is always unity.
Example: Consider a nilpotent matrix of order 2
A = [ 2 − 1, 4 − 2 ]
A 2 = [ 2 − 1, 4 − 2 ] [ 2 − 1, 4 − 2 ] = [ 0 0, 0 0 ]
So here :
A + I = [ 3 − 1 4 − 1 ] [ 3 − 1 4 − 1 ] ⇒ |A + I| = 1
Calculation: Given A is a 10 × 10 matrix and A5 is a null matrix, So, A is a nilpotent matrix of order 10. Also given I is the 10 × 10 identity matrix. Then the determinant of A + I = 1 (unity