GATE Overflow Test Series | Mock GATE | Test 5 | Question: 24

Question

Let $P$ be a matrix of order $3 \times 3$ whose elements are real numbers. If $P^2 = 0$, then the eigenvalues of $P$ are ________

• A. $0,0,0$
• B. $0,0,1$
• C. $0,1,1$
• D. $1,1,1$

Comments

@gatecse Why option B is not correct 
P is Nilpotent matrix and A^2 = 0

since power is 2 then two eigen values must be zero 

then third can be 1 isn't it?

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Determinant and Trace of nilpotent matrix is zero
Due to trace , option A only satisfies

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thanks

Answer 1

Given, $P^2 = 0$

If $X_1 , X_2 , X_3$ are the eigenvalues of $P$, then the eigenvalues of $P^2$ are $X_1^2, X_2^2$ and $X_3^2$

But $P^2 = 0, \implies X_1^2 = X_2^2 = X_3^2=0.$

$X_1^2 = 0 \rightarrow X_1 = 0$

$X_2^2 = 0 \rightarrow X_2 = 0$

$X_3^2 = 0 \rightarrow X_3 = 0$

Comments

p^2 ==0
is idempotent matrix
lamda can be 0 for all or 1 for all

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why not nilpotent? Why not C) ans?

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@akhil0699 idempotent is $A^2 = A$

$P$ is nilpotent, and eigen values of a nilpotent matrix is always $0$ which is also proved in the selected answer.