[PDF] EXERCISES OF MATRICES OPERATIONS Question 1

Two classical theorems on commuting matrices

AM= MA Then either M = ° or M is nonsingular Furthermore if A = A (so that M commutes with each element of ~l) then M is scalar :l PROOF Suppose that the rank of M is r, and write (II" M = P ° where P, Q are nonsingular and II' is the r X r identity matrix Then for each AE~l, (2) (II' (P-IAP) ° ° 0) _ (QAQ-I) Put (All P-IAP= A~I

EXERCISES OF MATRICES OPERATIONS Question 1

EXERCISES OF MATRICES OPERATIONS 3 (24) If A,Bare both n×nmatrices and ABis singular, then Ais singular or Bis singular (25) Ais diagonalizable, then Ais non-singular (26) Ais symmetric, then Ais non-singular

2 ALGEBRE` - Major-Prépa

6 La matrice triangulaire A admet deux valeurs propres r´eelles 1 et −1 : elle est diagonalisable a) Si M ∈ R, alors AM = M3 = MA et M ∈ C Par la question pr´ec´edente, M est donc de la forme αI + βA b) Comme A2 = I, on a (αI +βA)2 = (α2 +β2)I +2αβA et cette matrice vaut A lorsque α2 + β2 = 0 et 2αβ = 1, on a donc β

TD 03 : Matrices

(c)Montrer qu’il existe une base B de R3 dans laquelle la matrice de u est égale à 0 1 0 0 0 0 0 0 2 (d)Déterminer les endomorphismes qui commutent avec u 6 / Montrer que 0 0 0 1 1 −1 2 2 −2 est semblable à −1 0 0 0 0 0 0 0 0 7 / Montrer qu’une matrice A ∈M n(K) est semblable à la matrice dont

Rank of Matrix by Determinant

Example 1 Find the Rank of Matrix using Determinant ????= 1 2 3 2 4 7 3 6 10 ????=140−42−220−21+312−12 =1−2−2−1+30

Some Linear Algebra Notes

ij] is row (column) equivalent to a unique ma-trix in reduced (column) row echelon form The uniqueness proof is involved, see Ho man and Kunze, Linear Algebra, 2nd ed Note: the row echelon form of a matrix is not unique Why? Theorem 2 3 Let Ax= band Cx= dbe two linear systems, each of mequations in nunknowns If

Rank of Matrix by Normal Form - WordPresscom

R3 R3 –R2, ????≅ 1 0 0 2 0 0 3 1 0 C2 C2-2C1, ????≅ 1 0 0 0 0 0 0 1 0 As, Normal Form of given matrix A is having Identity Matrix of Order 2 rank (A)= r(A) = 2 ????≅ 1

64 Hermitian Matrices - Naval Postgraduate School

Ch 6: Eigenvalues 6 4 Hermitian Matrices We consider matrices with complex entries (a i;j 2C) versus real entries (a i;j 2R) 1 in R the length of a real number xis jxj= the length from the origin to the number

TD - Matrices

Exercice 17 Soit A ∈Mn (R)une matrice nilpotente d’ordre p >1 On pose B =In −A 1 Montrer que B est inversible et exprimer son inverse à l’aide de A (penser à la factorisation de I −Ap) 2 Application : B = 1 −1 0 0 0 1 −1 0 0 0 1 −1 0 0 0 1

Matrix Multiplication - University of Plymouth

Table of Contents 1 Introduction 2 Matrix Multiplication 1 3 Matrix Multiplication 2 4 The Identity Matrix 5 Quiz on Matrix Multiplication Solutions to Exercises

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EXERCISES OF MATRICES OPERATIONS

Throughout, we assume that the dimensions of the matrices in this note make sense. Question 1.Which of the following statements must be true? (1) IfA2=A, thenAmust be either the identity matrix or the zero matrix. (2) IfAis a 2×2 matrix and|A|= 4, then|2A|= 8. (3) IfAT=-A, then|A|= 0. (4) IfA2=I, thenA=IorA=-I. (5) IfA2= 0, thenA= 0. (6) IfAB= 0, thenA= 0 orB= 0. (7) If A is ann×nmatrix andAn= 0, thenA= 0. (8) IfAis symmetric, thenATis symmetric. (9) IfAis symmetric, then-Ais skew-symmetric. (10) IfA,Bis symmetric, thenABis symmetric. 1

2 EXERCISES OF MATRICES OPERATIONS

(11) IfA,Bis symmetric, thenA+Bis symmetric. (12) IfAC=BC, thenA=B. (13) IfCA=CB, thenA=B. (14) IfCis invertible andAC=BC, thenA=B. (15) IfAB=In, then so isBA. (16) IfA,Bare bothn×nmatrices andAB=In, then so isBA. (17) The nullity ofAis the same as the nullity ofAT. (18) BothA,Bare invertible, then so isA+B. (19) BothA,Bare invertible, then so isAB. (20) IfA,Bare bothn×nmatrices andABis invertible, then bothA,B are invertible. (21) BothA,Bare singular, then so isA+B. (22) BothA,Bare singular, then so isAB. (23) IfA,Bare bothn×nmatrices andABis singular, then bothA,B are singular.

EXERCISES OF MATRICES OPERATIONS 3

(24) IfA,Bare bothn×nmatrices andABis singular, thenAis singular orBis singular. (25)Ais diagonalizable, thenAis non-singular. (26)Ais symmetric, thenAis non-singular. (27) If all the eigenvalues ofAare 1, thenAis similar to the identity matrix. (28) If all the eigenvalues ofAare 1, thenAis non-singular. (29) IfAis invertible, thenA2is invertible. (30) IfAis invertible, thenAATis invertible. (31) IfAis invertible, thenATis invertible. Question 2.IfAis row equivalent toB, then which of the following statements must be true? (1) Ifλis an eigenvalue ofA, thenλis also an eigenvalue ofB. (2)Acan be obtained fromBby a finite step of elementary row opera- tions. (3)AX= 0 andBX= 0 has the same solutions.

4 EXERCISES OF MATRICES OPERATIONS

(4)AX=bandBX=bhas the same solutions for anyb. (5) rank(A) = rank(B). (6)A,Bhave the same reduced row echelon form. (7)ACandBCare row equivalent for anyC.quotesdbs_dbs16.pdfusesText_22