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VED1
E L-LOVELY-H math18-1 IInd 21-10-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 VIth 10-9-12

fJekJh^18 L t:{j (w?fNqe;) L noE ns/ gqeko

t:{j :'r d/ y/soh r[D (Field properties of Matrix Addition) B'N

(i) d' w?fNqe; fJZe jh eqw d/ (Order) iK nB[ebB (Conformable) jB sK fJBQK d/ i'V s/ gqkgs
w?fNqe; th fJ;h eqw dk j[zdk j?.
i/eo A = [a ij ] m × n ; B = [b ij ] m × n j't/ sK

A + B = C ⇒ [a ij b ij ] m × n ; :'r dk ;zuoD r[D (Closure Law)
(ii) i/eo d' w?fNqe; nB[ebB (;wkB^eqw d/) jB, sK
A + B = B + A :'r dk eqw^ftfBw/sk r[D (Commutative Law for Addition)

A + B = [a ij b ij ] m × n = [b ij + a ij ] m × n = B + A j't/ sK nB[ebB w?fNqe; dk :'r, eqw ftfBw/ r[D
dk gkbD eod/ jB.
(iii) i/eo fszB w?fNqe; A, B, C nB[ebB jB, sK

(A + B) + C = A + (B + C) :'r dk ;fjuo r[D (Associative Low for Addition)

(iv) i/eo A fJZe w?fNqe; m × n eqw dk j? ns/ 0 (Iho') w?fNqe; th T[;h eqw dk j? sK A + 0 = A
fiZE/, 0 (Iho') w?fNqe; dk :'r ss;we ;zxNe (Identity Element) j?.
(v) i/eo A fJZe w?fNqe; m × n eqw dk j? ns/ A + (–A) = 0 j't/, sK –A w?fNqe; A w?fNqe; dk
:'r t:{seqw (Additive Inverse) j[zdk j?.
d' w?fNqe;K dk nzso (Subtraction of Two Matrices)
A = [a ij ] m × n ; B = [b ij ] m × n sK d' w?fNqe;K dk nzso A – B, T[j w?fNqe; C j? fi; ftZu i/eo C =
[c ij ] m × n i/eo c ij = a ij = b ij
Gkt c ij = a ij + (–b ij )

d' w?fNqe;K dk nzsol A ns/ B d/ :'iBkswe gqshb'w (Additive Inverse) Gkt A – B i'VB
s/ fwbdk j?.

18H6 w?fNqe; r[DB (Matrix Multiplication)

i/eo A = [a ij ], m × n eqw dk w?fNqe; j? ns/ B = [b ik ], n × p eqw dk w?fNqe; j?, T[d'A A ns/ B
w?fNqe;K dk r[DB AB = C; fiZE/, C = [c ik ], m × n eqw dk w?fNqe; A j?. Gkt i/eo d' w?fNqe; A
ns/ B nB[ebB j'D fiBQK ftZu A ;szGK dh ;zfynk B dhnK gzeshnK d/ pokpo j't/, sK d'jK
w?fNqe;K B{z r[Dk eo ;ed/ jB ns/ r[DB B{z AB d[nkok gqdofPs eod/ jB.
fJ; soQK w?fNqe; r[DB AB d/ ftnkge gd c ik B{z gqkgs eoB dh ftXh fJj j?^
A dh ithA gzesh ns/ B d/ kt/A ;szG d/ ;zrs oueK (;zxNeK) dh r[Dk eoBk s/ fJBQK dk :'rcb
b?Dk. fJ;B{z gzesh r[fDs ;szG ftXh efjzd/ jB.

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