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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= 161) lo ‘lw. '
USERSL m. mass 90.05001“ M 31sz 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: ‘=>‘ $9“! A ‘1‘: WMTNUE A=€\E‘L“'Ek ”"5“ EH mashin MA: &&C€\EI_..E‘)= ng'MQ...MEK#O
stow £19476}, =A,\ 0?. info.
3» MA 1e o. 2:‘ Sm MA+Q "men A": ELKQJQA .9: in “madam,
A Is liueﬁxgua. Dr. Doom Lin Alg MATH1850/2050 Section 2.3 Page 4 of 7 Theorem 2.3.4 If A and B are square matrices of same size, then det(AB) = det(A)det(B). Ang) = M (E‘s: 1:; R s) mus wen? AVE 1 CASES a
G) A is muemué "’ =3.“ (9 A is Sm'euut. aw [L ecurains M W 0’55 099' °‘ “'5' me o. ”Cm: MLei‘e;‘...E;‘ ?>) = Mcef‘)ML€;‘).~ oucCE'JHﬂ by (MA)
= awcLeﬂesﬂ er) M9:
= (MA(Mg.
CA$€® M93): METM 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 MAJ114160 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...,Qm60 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 MAMK‘=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 rewrite 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 nontrivial
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 3Xi3k '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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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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