query: How would you prove that exchanging two rows of an $n \times n$ matrix $A$ multiplies its determinant by $-1$?
positive_passages.docid: 17017
positive_passages.text: Row swap changing sign of determinant
negative_passages.docid: 71604
negative_passages.text: Why is the determinant of the 0x0 matrix equal 1?
negative_passages.docid: 2271
negative_passages.text: Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
negative_passages.docid: 110200
negative_passages.text: Determinant of a Special Symmetric Matrix
negative_passages.docid: 54740
negative_passages.text: Proofs of Determinants of Block matrices
negative_passages.docid: 70381
negative_passages.text: Find the determinant of a symmetric matrix
negative_passages.docid: 146438
negative_passages.text: Determinant of a special skew-symmetric matrix
negative_passages.docid: 60231
negative_passages.text: Number of equivalence classes of matrices under switching rows and columns
negative_passages.docid: 98135
negative_passages.text: Show that the determinant of $A$ is equal to the product of its eigenvalues
negative_passages.docid: 87676
negative_passages.text: Why is determinant a multilinear function?
negative_passages.docid: 52564
negative_passages.text: $m\times n$ matrix with an even number of 1s in each row and column
negative_passages.docid: 57318
negative_passages.text: Property of the conjugate transpose matrix with inner product
negative_passages.docid: 15452
negative_passages.text: Proving determinant product rule combinatorially
negative_passages.docid: 151526
negative_passages.text: Determinant of a matrix with different cases for its entries
negative_passages.docid: 69980
negative_passages.text: Proving that if $A$ is an $n\times n$ orthogonal matrix, then $\det(A) = \pm1$
negative_passages.docid: 65158
negative_passages.text: Oddity: Determinant of Skew-Symmetric $n\times n$ Matrices
negative_passages.docid: 81770
negative_passages.text: Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant