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

GATE Electrical Engineering (EE) 2021 Solved Papers

Question : 38 of 65

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Let

{A}

be a

10 × 10

matrix such that

{A}^{5}

is a null matrix, and let

{I}

be the

10 × 10

identity matrix. The determinant of

{A}+ {I}

is___________.

Solution:

Given,

{A}^{5}=0

∴

A is a nilpotent matrix of index ' 5 '.

∴

eigen values of A are all 0 .

| {A}|=

product of eigen values

=0

| {A}|=0

Now,

|A+I|=|A|+|I|

=0+1=1

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