linear algebra - Integer matrices with determinant equal to $1 ...
2017年7月14日 · The diophantine equation is solvable if $\gcd\{c_{11},\dots,c_{1n}\}=1$ and in that case the solutions depend on $n-1$ parameters. An advantage of this method is that you …
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Determinant of Matrix with Solved Examples
2025年1月3日 · The determinant of an orthogonal matrix is either +1+1 or −1−1. This property arises from the fact that the determinant represents the scaling factor of the matrix transformation. Since orthogonal transformations preserve …
Determinant - Wikipedia
For instance, an orthogonal matrix with entries in R n represents an orthonormal basis in Euclidean space, and hence has determinant of ±1 (since all the …
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Why is the determinant of a symplectic matrix 1?
The determinant is a continuous function, and the set of symplectic matrices with invertible $A_1$ is dense in the set of all symplectic matrices. So if you've proven that it equals 1 for all …
The determinant of a permutation matrix P is 1 or −1 depending on whether P exchanges an even or odd number of rows. From these three properties we can deduce many others:
8.1: The Determinant Formula - Mathematics LibreTexts
2023年7月28日 · \[\det E^{i}_{j}=-1\, ,\] where the matrix \(E^{i}_{j}\) is the identity matrix with rows \(i\) and \(j\) swapped. It is a row swap elementary matrix. This implies another nice property of the determinant. If two rows of the matrix are …
Determinant of a Matrix - Math is Fun
The determinant helps us find the inverse of a matrix, tells us things about the matrix that are useful in systems of linear equations, calculus and more. Calculating the Determinant. First of …
Determinants: Definition - gatech.edu
Learn some ways to eyeball a matrix with zero determinant, and how to compute determinants of upper- and lower-triangular matrices. Learn the basic properties of the determinant, and how …
Math 21b: Determinants - Harvard University
The determinant of the inverse of an invertible matrix is the inverse of the determinant: det(A -1 ) = 1 / det(A) [6.2.6, page 265]. Similar matrices have the same determinant; that is, if S is …
