query: For a $2$ by $2$ matrix A, why does $A^3=0$ imply $A^2=0$?
positive_passages.docid: 151806
positive_passages.text: Prove that $A^k = 0 $ iff $A^2 = 0$
negative_passages.docid: 66451
negative_passages.text: In a ring, how do we prove that a * 0 = 0?
negative_passages.docid: 33891
negative_passages.text: If $\mathbf A$ and $\mathbf B$ are Hermitian, when does $\det(\mathbf A-\lambda\mathbf B)=0$ have only real roots?
negative_passages.docid: 10828
negative_passages.text: If $A^2 = I$, then $A$ is diagonalizable, and is $I$ if $1$ is its only eigenvalue
negative_passages.docid: 139
negative_passages.text: Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
negative_passages.docid: 32103
negative_passages.text: Show that for a $2\times 2$ matrix $A^2=0$
negative_passages.docid: 152831
negative_passages.text: When does a complex matrix have a square root?
negative_passages.docid: 129743
negative_passages.text: $A$ be a $2\times 2$ real matrix with trace $2$ and determinant $-3$
negative_passages.docid: 120407
negative_passages.text: Is every noninvertible matrix a zero divisor?
negative_passages.docid: 119387
negative_passages.text: If $A$ is any matrix then $A^*A$ and $AA^*$ are Hermitian with non-negative eigenvalues
negative_passages.docid: 129583
negative_passages.text: Find 3x3 real matrix A such that $A^2 \neq I$ and $A^4 = I$, where I is identity matrix.
negative_passages.docid: 42334
negative_passages.text: Does $\lambda$ $\mathbf u$ $= 0$ imply $\lambda = 0$ or $\mathbf u$ $= 0$?
negative_passages.docid: 146243
negative_passages.text: Does there exist a matrix $\mathbf{A}\in\mathbb{R}^{3\times3}$ such that $\mathbf{A}^{2}=-\mathbf{I}$?
negative_passages.docid: 123726
negative_passages.text: $2\times 2 $ matrices over $\mathbb{C}$ that satisfy $\mathrm A^3=\mathrm A$
negative_passages.docid: 115296
negative_passages.text: Under what conditions does a matrix $A$ have a square root?
negative_passages.docid: 140252
negative_passages.text: Matrices - Conditions for $AB+BA=0$
negative_passages.docid: 4443
negative_passages.text: Why is the determinant zero iff the column vectors are linearly dependent?