Page 286 - DECO403_MATHEMATICS_FOR_ECONOMISTS_PUNJABI

VED1
E L-LOVELY-H math19-1 IInd 6-8-11 IIIrd 24-1-12 IVth 21-4-12 VIth 10-9-12

noEPk;soh dk rfDs

B'N T[dkjoD ti'A, i/eo
 15 

 13 7   
A =   T[d'A A′ =  32


 52 4    74 

;gPN j? fe ;zxNe A ij , fijV/ w?fNqe; A dh ithA gzesh ns/ jt/A ;szG ftZu ;h, gfotos
w?fNqe; A′ dh jthA gzesh ns/ it/A ;szG T[Zs/ oj/rk. fJ; bJh i/eo
′
A = [a ij ] ns/ A′ = [a ij ] ⇒ a a= ij
ij
B'N^i/eo n;hA A′ dk gfotos btKr/, sK fco s'A A gqkgs j' ikJ/rk.
) =
∴ ()A′′ = (A T T [A tt ] = A
′;
fJ;h soQK [KA ′ ]= KA fJZE/ K fJZe nkfdP okPh (Scalar) j?.
gqw/: 1^ i/eo A ns/ B d'B'A m × n eqw d/ w?fNqe; jB, T[d'A
( + )AB ′ = A′ + B′

T[gZsh^;kB{z gsk j? fe A ns/ B :'r d/ nB[ebB j'Dr/ i/eo d'B'A fJZe jh eqw m × n d/ j'D.
fJ; bJh wzB fbU.

A = [a ij ] ns/ B = [b ij ] T[d'A, C = A + B = [c ij ], fiZE/, c ji = a ji + b ij
j[D A′ = [ ]a = [ B′ = ]b , fiZE/ a ji = a ij ns/ b ji = b ij
ji ji
=
=
+
(A B+ )′ = C′ = [c ][a + b ][a ] [b ] = A′ + B. ′
ji ij ij ji ji

19H2 d' w?fNqe; d/ r[DB d/ gfotos
(Transpose of Product of Two Matrices)
gqw/: 2^i/eo w?fNqe; A ns/ B r[DB d/ bJh nB[ebB jB, sK AB dk gfotos T[seqw ftZu
gfotosK d/ r[DBcb d/ pokpo j'J/rk Gkt i/eo A ns/ B eqw nB[;ko m × n ns/ n × p eqw
d/ jB, T[d'A ( )AB ′ = B A ′′
T[gZsh^AB dk eqw m × p j?, sK ( )AB ′ eqw p × m dk j't/rk, B′ dk eqw p × n ns/ A′ dk
′
eqw n × m dk j?, fJ; bJh BA ′′ , p × m eqw dk j'J/rk. Gkt d'B'A ( )AB ′ ns/ B .A′ pokpo
eqw d/ jB.
wzB fbU T[d'A A = [a ij ], B = [b jk ], T[d'A

( )AB ′ dk (k – i)tK ;zxNe & (AB) dk (i – k)tK ;zxNe
n
= ∑ ab
jk
ij
j= 1
′
()′ = (b A′ = (a ) fiZE/ b ′ = b ′ ,a ′ = a
B
)
kj ji jk jk ji ij
n
fJ; soQK BA ′′ dk (k – i)tK ;zxNe = ∑ ab
jk ij
j= 1
n
jk ij ∑
= ∑ a b = n ab
ij
jk
j= 1 j= 1
Gkt BA ′ dk (k – i)tK ;zxNe & ( )AB ′ dk (k – i)tK ;zxNe

280 LOVELY PROFESSIONAL UNIVERSITY