Linear algebra

Question

A² − A = 0, where A is a 9×9 matrix. Then

(a) A must be a zero matrix

(b) A is an identity matrix

(c) rank of A is 1 or 0

(d) A is diagonalizable

Comments

i am getting, option C.

option B and option D you can eliminate easily,

For every Zero matrix it is true ( Note that it's rank is 0 )...

but there are some other matrices which the given statement is true. ( make only one element in any row is 1 and remaining all elements are 0 ===> rank of this matrix is 1 )

ex 1) make 1^st element of 1^st row is 1 and take remaining elements are 0

ex 2) make 3^rd element of 1^st row is 1 and take remaining elements are 0 ..... etc

 

But note that, every matrix which have rank = 1 , doesn't need to satisfy the question.

Option C conveying, which matrix satisfies the question, it have rank=1 or 0.