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# 2.3 - Dr Doom Lin Alg MATHl850/2050 Section 2.3 Page 1 of 7...

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Unformatted text preview: Dr. Doom Lin Alg MATHl850/2050 Section 2.3 Page 1 of 7 Chapter 2 - Determinants Section 2.3 - Properties of the Determinant Function Basic Properties of Determinants: Theorem If A is a square matrix and k is any scalar, then det(kA) = k”det(A). Example: Say A is a 4 X 4 matrix with det(A) = —2, and k = 3, then MC“): 3“-&.&A= 8\-c—z)=—lez. Example: With k = 3 and A = [ j: 1:? J , MA: K4145- M (2, M: 314.17% 3- 7a ~47. In general it is not true that det(A + B) = det(A) + det(B). Example: An instance Where det(A + B) 55 det(A) + det(B) , _ 1—2 _ 0 3 A=o .mgso WlthA_[0 0],B_[0 2], «Li , A+s=[‘ '\ ﬂ(A&3;1¢O=&AA+M£. O I Dr. Doom Lin Alg MATHl850/2050 Section 2.3 Page 2 of 7 Theorem 2.3.1 Let A, B and C’ be n X n matrices that differ only in a single row, say the TM and assume that the TM row of C’ can be obtained by adding the corresponding entries in the TM rows of A and B. Then det(C’) = det(A) + det(B). NOTE: The same result holds for columns. 2 3 —1 2 3 —1 Example: WithA= , = (0 2 0) ,and 3 5 —2 —3 5 3 b 1.11: 2C 1mm -\\=—z C: 0 2 , n :1, —1_ . 0 10 1 1M)— “Wu—imp —;\=4. 1H. 1 _\ oUCC= 1-61) lo ‘lw. ' USERSL m. mass 90.05001“ M 3-1sz Qwerty.- So 1.1a: «Mm 1.1g. admin: mime, If A and B are square matrices of same size, then det(AB) = det(A)det(B) Example: WithA= [4 2 3] B=[_Z —§], aed‘-i‘\="\0>‘e'g‘i‘get'f Nb- {‘3‘ ﬂ M®)=—uo 1&1 sue, But in order to prove this theorem we need some preliminary results, includ- ing the following Lemma Dr. Doom Lin Alg MATH1850/2050 Section 2.3 Page 3 of 7 Lemma 2.3.2 If B is an n X 71 matrix and E is an n X n elementary matrix, then det(EB) det(E)det(B Proof of Lemma: 6A1 E \'\$ 0%“;qu FEM I“ (if! WEMHMG 9.600 00 QCQRS % —’9' Eis M663 = -4M:&.Q-‘mn maniac. Tum) 2;,6925 Repeated applications of this Lemma gives us that det(E7....E2E1B) = det(E7.) . . . det(E2)det(E1)det(B). Theorem 2.3.3 Determinant Test for Invertibility A square matrix A is [email protected](A) 7é 0. ii: Auo ouw it - . I Proof: ‘=>‘ \$9“! A ‘1‘: WMTNUE- A=€\E‘L“'Ek- ”"5“ EH mashin- MA: &&C€\EI_..E‘)= ng'MQ-...MEK#O stow £19476}, =A,-\ 0?. info. 3» MA 1e o. 2:‘ Sm MA+Q "men A": ELKQJQA .9: in “madam, A Is liueﬁxgua. Dr. Doom Lin Alg MATH1850/2050 Section 2.3 Page 4 of 7 Theorem 2.3.4 If A and B are square matrices of same size, then det(AB) = det(A)det(B). Ang) = M (E‘s: 1:; R s) mus wen? AV-E 1 CASES a G) A is muemué "’ =3.“ (9 A is Sm'euut. aw [L ecu-rains M W 0’55 099' °‘ “'5' me o. ”Cm: MLei‘e;‘...E;‘ ?>) = Mcef‘)ML€;‘)---.~ oucCE'JHﬂ by (MA) = awcLeﬂesﬂ er) M9: = (MA-(Mg. CA\$€® M93): MET-M E;‘-._.-J.9za‘ MUS) w: rum. Jt W Q m 47.045 = o 0% A = 0 (Tan 0N Vaméos PAGE) 5. @Q~M\$=O = ﬂew Dr. Doom Lin Alg MATHl850/2050 Section 2.3 Page 5 of 7 One very nice consequence of the theory developed here is that we can ﬁnally tackle the proofs of a couple of theorems from earlier sections: Proof of Theorem 1.6.5. If A and B are square matrices of the same size and if AB is invertible, then so are A and B. AQ> is uivaniue «K MA?) #0 Q MAJ-114160 L“ \$14408. £9,4st LQK A g g ALE liMezfusLE. Proof of Theorem 1.7.1. (c) Properties of Triangular Matrices A triangular matrix is invertible iff its diagonal entries are all non—zero. A \‘s A, m'vcurnue @‘C MAH‘O ~« 4\‘Qn_-.qm\#0 4K 0.“,Qw...,Qm-60 LMM Dr. Doom Lin Alg MATHl850/2050 Section 2.3 Page 6 of 7 Theorem 2.3.5 If A is invertible, then 1 det(A) det(A_1) = _ -\_ _\ Example: WithA=[ 3—22] MA=-1. 5°chsz - :— Example: Say B a 4 X 4 matrix with det(B)=3. Then Au(o.e.-)=;1§¢T/5, [email protected]‘=M(ﬁ%"l=llzl‘ﬁz= Proof of 171771.235: AA“ —— so MGM" \ so MA-MK‘=A JJK‘W. This is the theorem that keeps on growing, new edition: Theorem 2.3.6. Equivalent Statements If A is an n X 71 matrix, then the following statements are equivalent, that is, they are all true or all false. (a) A is invertible. (b) Ax = 0 has only the trivial solution. (c) The reduced row echelon form of A is [7,. (d) A is expressible as a product of elementary matrices. (e) Ax— — b 1s consistent for every n X 1 matrix b. f Ax— — Ab has exactly one solution for every n X 1 matrix b. Dr. Doom Lin Alg MATHl850/2050 Section 2.3 Page 7 of 7 Linear Systems of the form AX = AX This is a brief intro to what we will study in detail in Chapter 7. We are interested in linear systems of n equations in n unknowns of the form AX=AX 61—9 X}~Az=g which we can re-write as a homogeneous system (AI — A)X = 0. Of interest here is determining the values of A for which the system has a nontrivial solution; such a value is called an eigenvalue. If A is an eigenvalue of A, then the nontrivial solutions of AX = AX are called the eigenvectors of A corresponding to A. It follows from Thm 2.3.3 that the system (AI — A)X = 0 has non-trivial solutions iff detox] — A) = 0. This equation is called the characteristic equation of A. We ﬁnd A by solving this equation. Example: m (A ’ [ t 3Xi3-k '5 {Hi/1*— Sig] Sn .)\-—'S is Au thehd/KUE 2' u “L R 2a ‘L 7, . > X faikrsAcqwqwue Q EWL- ...
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