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Unformatted text preview: Dr. Doom Lin Alg MATH1850/2050 Section 2.2 Page 1 of 5 Chapter 2  Determinants
Section 2.2  Determinants by Row Reduction In this section we are trying to come up with a more efﬁcient method for calculating deter—
minants. Theorem 2.2.1. Let A be a square matrix. lf A has a row (or column) consisting entirely
of zeros, then det(A) = 0. Proof: ‘80“ Common. emu» Hume we am) (comm) op ms
0%.} A: 0+0*...+O=o. Theorem 2.2.2. Let A be a square matrix. Then det(AT) = det(A). Proof:
‘U‘A " clinch ’f Que—61. *   +Q—c“c.‘m Mao; we Lli Raul r' f F ~ .
M AT= 0g ZniQiaa’ﬁ 'l‘ _ _ Quack" Acme Tue Lﬂ‘ moms. Dr. Doom Lin Alg MATHl850/2050 Section 2.2 Page 2 of 5 Elementary Row Operations Theorem 2.2.3. Let A be an n X 71 matrix. (a) lf B is the matrix resulting when a single row (or single column) of A is multiplied
by a scalar k}, then det(B) = k;  det(A). (b) lf B is the matrix that results when two rows (or two columns) of A are interchanged,
then det(B) = —det(A). (c) If B is the matrix that results when a multiple of one row (or one column) is added to
another row (column, respectively) of A, then det(B) = det(A). 2 —1 __3n 7. —) _
Examples: WithA= 0 3 ,and MA:— 1 (I) o 3 "1/~
—1 —1
2 —1 0
(i) A—eBF 0 3 0 so M%=3J&A=—3\1=—3c.
2903) 3 3 —6
2 —1 0 % MA \
(ii) _’> B: —i —i 2 o m =— =1.
919393 2 0 3 0 s
0 —3 4 (iii)A———>B3= 0 3 0 ScabieacélaﬁA’Q. Dr. Doom Lin Alg MATH1850/2050 Section 2.2 Page 3 of 5 Elementary Matrices The following is just Thm. 2.2.3 specialized to the case when/{is
an elementary matrix. X. A =1}. 5 MIW=L Theorem 2.2.4. Let E be an n X n elementary matrix. (a) lf E results from multiplying a row of In by a scalar k}, then det(E) = k}.
(b) If E results from interchanging two rows of I”, then det(E) = —1.
(c) lf E results from adding a multiple of one row of In to another, then det(E) = 1. Examples: WithE1=[1 0], E2=[ 0 1],E3= Met7‘1 Met:L Theorem 2.2.5. If A is a square matrix with two proportional rows or two proportional
columns, then det(A) = 0. Proof: SAVAHAS z. Vmohzrdwm. laws, 5,5 5», Qi=k25 mtg hem
A ——> [5 w omno (Wm) is Awe: 7.629s)
261.7%: 3
g MumA (mm. «mm
Se. 43:560. Example: Introducing Zero Rows[Columns] 1—125 A 3 0 1 —3 mg (13:1[1‘50 ﬂit/AWO
' 2 —2 4 10
0 5 1 —1 \ \ 7. s
, 3 ° ‘0‘: =3 Msao
(2113—22. 2 g L _‘ X “hang go Dr. Doom Lin Alg MATH1850/2050 Section 2.2 Page 4 of 5 Theorem 2.1.3. If A is an n X n triangular matrix (upper, lower, or diagonal) then det(A)
is the product of the entries on the diagonal. That is = allagg . . . am. Proof: The proof suggested by the textbook uses cofactor expansion to calculate the deter—
minant. ' 4»; Qu
A. 2:21:32: MA= a“a... aw o 0...Qm o "QM = ennui‘3‘“ :12 11:21: a . .Qnu 2 QuQu'  Ian As an alternative, we can use the deﬁnition of determinants from Section 2.4. (determinant
as the sum of signed elementary products)
yd stanza ecu‘7 we.me w Vt! \ 0? “men me 25m aLkA‘ "' Q\\qn"'am~ . _ _ . . n
K’ Aisocwte» 96¢.HDTRTIOO (‘22)33"2‘A'3 “’4‘” l‘ We” (MS 0 ngwo Q D1“. Doom Lin Alg MATH1850/2050 Section 2.2 Page 5 of 5 Example: Using Row Reduction to Evaluate Determinants 1—1 2—1 _ 2—21—3
A“ —1 1 4 6 on. use \A\ 0 1 2 —1 o 34" lute«1'15. 3 Example: Row Operations and Cofactor Expansion 1—1 2—1
2—2 1—3
A: —1 1 4 6
0 1 2—1 _ 'L 0 \—3 «+2.
MA " 5‘ o “L = 4 ('4) '2, \ —3
cl 1\ —\ q 4
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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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