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MATRICES - UNB

If A and Bare square matrices such that [AB=BA=II (22) then B is ca II ed the inverse of A and is denoted B = A -I (or equivalently A is called the inverse of Band is denoted A= B-1) Only nonsingular square matrices have an inverse Singular matrices do not have an inverse Given the matrices A = 2 -2 2 5 -4 3 7 -5

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(C1) The only matrices in Mn(F)commuting with all matrices in S are the scalar matrices (C2) The only subspaces of the column space V = Fn that are invariant under the action of all matrices in Sare0 and V Note 2 2 In the above notation, if F is a subfield of K then dimFF S= dimKK S,soF S= Mn(F)if and only if K S= Mn(K) Theorem 2 3

Quizz 18 Matrices - WordPresscom

Commentaire sur les matrices de rang un et l’exercice 9 Une matrice M de rang un peut s’écrire M ˘XYT avec X et Y non nuls dans Kn « Par associativité » du produit que M2 ˘YTXM ˘tr(M)M (car YTX est une matrice scalaire, ie appartenant à M1,1(K)) En particulier, si tr(M) ˘ 0, alors M

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