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Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable then
  • a)
    A is nilpotent
  • b)
    A is not nilpotent
  • c)
    The minimal polynomial of A is linear
  • d)
    A has a non-zero eigenvalue.
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable the...
Explanation:

To prove that the correct answer is option 'A', let's analyze each option one by one.

a) A is nilpotent:
A matrix A is nilpotent if there exists a positive integer k such that A^k = 0, where 0 is the zero matrix.
Since A is a 2 x 2 real matrix of rank 1, it means that the column space and row space of A are one-dimensional. This implies that A has only one non-zero eigenvalue, say λ.
If A is nilpotent, then all eigenvalues must be zero. Since A has a non-zero eigenvalue, it cannot be nilpotent. Therefore, option 'a' is incorrect.

b) A is not nilpotent:
Based on the explanation in option 'a', A is not nilpotent. Therefore, option 'b' is incorrect.

c) The minimal polynomial of A is linear:
The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates the matrix. In other words, it is the polynomial p(x) such that p(A) = 0.
Since A is a 2 x 2 real matrix of rank 1, it means that A has only one non-zero eigenvalue, say λ. The minimal polynomial of A will be a polynomial of degree 1, i.e., linear, with λ as its root. Therefore, option 'c' is incorrect.

d) A has a non-zero eigenvalue:
Since A is a 2 x 2 real matrix of rank 1, it means that A has only one non-zero eigenvalue, say λ. Therefore, option 'd' is correct.

To summarize, the correct answer is option 'A' - A is nilpotent.
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Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable the...
Option B is right
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Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable thena)A is nilpotentb)A is not nilpotentc)The minimal polynomial of A is lineard)A has a non-zero eigenvalue.Correct answer is option 'A'. Can you explain this answer?
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