GATE Electrical 2021 | Question: 38

Question

Let $A$ be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let $I$ be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is ___________________.

Answer 1

ans 1

A is null matrix

then A^5=A

A+I=I

where is I identity matrix

determinant of (A+I)=determinant(I)

determinant of I=1(ans)

Comments

if A^5 = null matrix then how  A^5 = A?

 

Answer 2

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