Today while solving some Linear algebra exercises, one question came to my mind
Question: suppose given $n×n$ matrix $A$ has nullity $2$ then what can you say about geometric multiplicity (GM) and algebraic multiplicity (AM) of eigenvalue $0$?
My attempt: Since given that $nullity(A)>0$ that imply $0$ must be an eigenvalue of matrix $A$.
Since nullity of $A$ is dimension of nullspace of $A$ i.e. dimension of solution space of of homogeneous system $Ax=\bar{0}$. Hence, Geometric multiplicity(GM) of eigenvalue $0$ i.e. $GM(0)=nullity(A)=2$? (Is am I right)
and as we know $GM≤ AM$ for every eigenvalue of matrix, hence here we have, $2≤AM(0)$, so $AM(0) ≥2$. Can we say $AM(0)=2$ exactly?