query: Square matrix A is singular $\iff \operatorname{rank}(A)<n$?
positive_passages.docid: 27474
positive_passages.text: A square matrix $n \times n$ is an invertible matrix iff the rank of the matrix is $n$.
negative_passages.docid: 41026
negative_passages.text: How to prove and interpret $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$?
negative_passages.docid: 41932
negative_passages.text: Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular.
negative_passages.docid: 29788
negative_passages.text: Eigenvalues of a matrix whose square equals its transpose
negative_passages.docid: 97487
negative_passages.text: How to prove whether a matrix has rank $1$
negative_passages.docid: 89729
negative_passages.text: If $X$ is a singular matrix, then is $X'X$ singular?
negative_passages.docid: 87387
negative_passages.text: Linear equations given by $Ax = b$ have a solution if and only if $\operatorname{rank}(A|b) = \operatorname{rank}(A)$
negative_passages.docid: 149219
negative_passages.text: If $A$ is a non-square matrix with orthonormal columns, what is $A^+$?
negative_passages.docid: 134402
negative_passages.text: $\mathrm{rank}(A)+\mathrm{rank}(I-A)=n$ for $A$ idempotent matrix
negative_passages.docid: 27248
negative_passages.text: If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$
negative_passages.docid: 46128
negative_passages.text: Representation of Matrix with Rank 1
negative_passages.docid: 123665
negative_passages.text: A question in Subspaces in linear algebra
negative_passages.docid: 109167
negative_passages.text: Nth roots of square matrices
negative_passages.docid: 94639
negative_passages.text: Proof that the rank of a skew-symmetric matrix is at least $2$
negative_passages.docid: 119435
negative_passages.text: Rank of the difference of matrices
negative_passages.docid: 82688
negative_passages.text: Least rank of a matrix
negative_passages.docid: 127591
negative_passages.text: How to prove that the collection of rank-k matrices forms a closed subset of the space of matrices?