query: An $ n × n $ matrix $A$ satisfies $A^2 = 0 $. Can the rank of $A$ be $n$?
positive_passages.docid: 55820
positive_passages.text: If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.
negative_passages.docid: 86125
negative_passages.text: If $A$ is idempotent, then the eigenvalues of $A$ are $0$ or $1$
negative_passages.docid: 48660
negative_passages.text: Proof that determinant rank equals row/column rank
negative_passages.docid: 126711
negative_passages.text: What is the square matrix $A$ equal to if $A=A^2$
negative_passages.docid: 4952
negative_passages.text: Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$).
negative_passages.docid: 18297
negative_passages.text: Suppose that each of the row sums of an $n\times n$ matrix $A$ is equal to zero. Show that $A$ must be singular.
negative_passages.docid: 44383
negative_passages.text: Finding rank of a matrix depending on parameter a
negative_passages.docid: 41411
negative_passages.text: How to show that the nth power of a $n \times n$ nilpotent matrix equals to zero $A^n=0$
negative_passages.docid: 66729
negative_passages.text: For a square matrix $A$ presence of $n$ linearly independent eigenvectors implies full rank, thereby invertible?
negative_passages.docid: 30866
negative_passages.text: Relation between rank and number of distinct eigenvalues of a matrix
negative_passages.docid: 41514
negative_passages.text: Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
negative_passages.docid: 61274
negative_passages.text: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
negative_passages.docid: 142642
negative_passages.text: $A=\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix}$, $X^2=A$ has no solutions
negative_passages.docid: 15202
negative_passages.text: the root of $N$th transpose matrix
negative_passages.docid: 69606
negative_passages.text: How to prove rank(A+) is no more than rank(A)?
negative_passages.docid: 128453
negative_passages.text: If $A$ has two eigenvalues $\lambda _1, \lambda_2$ and $\dim (E_{\lambda_1})=n-1$, then $A$ is diagonalizable
negative_passages.docid: 76200
negative_passages.text: Is the rank of $AB$ always equal to the rank of $BA$?