Determinants & Inverse Matrices
A matrix has an inverse exactly when its determinant is not equal to 0 ***** *** 2⇥2inverses Suppose that the determinant of the 2⇥2matrix ab cd does not equal 0 Then the matrix has an inverse, and it can be found using the formula ab cd 1 = 1 det ab cd d b ca Notice that in the above formula we are allowed to divide by the determi-
Determinants, Matrix Norms, Inverse Mapping Theorem
determinant as you perform these row operations, and you will have calculated detA (There are shortcuts for this procedure, but that’s another story We’ll mostly be dealing with 2 2 and 3 3 matrices, for which one can just use the explicit formulas above ) This observation is also the key to the main theoretical signi cance of the
Invertible matrix
that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A Properties Let A be a square n by n matrix over a field K (for example the field R of real numbers) Then
LECTURE 10: DETERMINANTS BY LAPLACE EXPANSION AND INVERSES BY
Algorithm (Laplace expansion) To compute the determinant of a square matrix, do the following (1) Choose any row or column of A (2) For each element A ij of this row or column, compute the associated cofactor Cij (3) Multiply each cofactor by the associated matrix entry A ij (4) The sum of these products is detA Example We nd the
12 Elementary Matrices and Determinants
Thus, the determinant of a diagonal matrix is just the product of its diagonal entries Since the identity matrix is diagonal with all diagonal entries equal to one, we have: detI= 1: We would like to use the determinant to decide whether a matrix is invertible or not Previously, we computed the inverse of a matrix by ap-plying row operations
Chapitre 3 : Déterminants
à la matrice In Or, d’après la proposition 3 ces opérations multiplient le déterminant par un nombre non nul Ainsi, à l’issu de ces opérations, il existe λ ∈K∗ tel que λdet(A)=det(In)=1 Ainsi, det(A)6= 0 ⇐= On raisonne par contraposée et on suppose que A n’est pas inversible Le rang de A, qui est la dimension
DETERMINANTS EXERCICES - bagbouton
1 Préciser le rang de la matrice A 2 Donner une condition nécessaire et suffisante pour que A soit la matrice d’un projecteur 3 On pose B A tr A I= −2 n Calculer le déterminant de B 4 Donner une condition nécessaire et suffisante pour que B soit inversible 5 Calculer B2 Calculer −1 dans le cas où B est inversible EXERCICE 10 :
Determinants and eigenvalues
The determinant of a triangular matrix is the product of its diagonal entries A = 123 4 056 7 008 9 0 0 0 10 det(A)=1· 5 · 8 · 10 = 400 facts about determinantsAmazing det A can be found by “expanding” along any rowor any column
Exercices16 Déterminants
Soit A 2M2n¯1(K) une matrice antisymétrique Prouver que A n’est pas inversible b Prouver l’existence de matrices antisymétriques inversibles dans M2n¯2(K) 2 Vuauconcours Soit E un K-ev de dimension finie n 2N⁄ et f 2L(E) Démontrer que dimE est paire dans les cas suivants : a f 2 ˘ ¡idE; b f 2GL(E) et 9g 2GL(E), f –g
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