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.
Let A be a 2 x 2 real matrix of rank 1. If A is not diagonalizable the...
Option B is right