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Re FOI I Ry |- ad-be an te No: of poe No: ap column et Expression [value He Determmmonrt ve presents ttt, E eteh ad * thy FO tal ott kh Cofactor & Minors ofanElement | Minors: Minors of an element is defined as the minor determinant obtained by deleting a particular row or column in which that element lies. 4 943 e.g. inthe determinant D = |421 422 423] minor of a, denotes as G34, 932 33 421 423 las My = | | and so on. & 431 33 ) ee ay" Gaps \ PE) ca \ \ Cofactor & Minors of an Element \ Cofactor : It has no separate identity and is related to the minors as C;; = (—1)'*/M,;, where 'i' denotes the row and ‘j' denotes the column. Expansion of Determinant The value of determinant is defined as the sum of the product of elements of any row (column) by their corresponding co-factors. &, Ga ay Go, G22 922 = QyCy + G2, +22 Sa (Akeng Ri) Cz, Gg G3, QyCy, +Gy,C2, +S, Sa) (Akeng a3 = \% Shy Q ate, Wb, 2 x Rhy Q | Properties of Determinants -_ @ “If each element of any row (or column) can be expressed as a sum of two | terms, then the determinant can be expressed as the sum of two determinants. Oy tee ites mC a by cy x ee 2 e.g. Agth bot P Cot¥] = JagtebotP cot |aarebs Pca a3 bg C3 az bg o3 az bs c3 P-5 =» <1 ay a2 )} \Qn bs ty \ \ Sa ey S& : 33] Jan ky oS Reverse of P-5 is summation of determinants. To acld two Determmnans * Hoey Ahold hove Same order He ARB corvesprding Peos|cofernns Shovid be Klenticad except Tre te Adldlition tare: pice oLorg He Ten idertico2 vows | cokuenn —~ Properties of Determinants a ' The value of a determinant is not altered by adding to the elements of any row | P-6 ™ (or column) the same multiples of the corresponding elements of any other row (or column). ay by Cy a,+maz, by+mbz c, +mc2 eg. LetD=J]a, bz cg] and D'= a2 by C2 az bz C3 a3+na, b3tnby cz +ncy _ Then D’ = D. (hale ce Gax, Tete nh aoe Yam Rar LR e 3 ee ( q Qn. ba Os, 3%, le-= bytes Ste Ie: by -| Ri Pith Py 2+ Re Gor, bath, 2+ =O Ge, ba fp a,b Q@y ba So G& by %& 4 ee bth S82 Betas, byt bo Ry Rae Ram RRL @ ba Sa \= Poa, baat S20) 1 & ba, ae | q a By using the operation Rj; > xk; + yRj+2RyG,k #i), the value of the determinant becomes x times the original one. While applying this property ATLEAST ONE ROW (OR COLUMN) must remain unchanged. Sum of product of elements of any row (column) with cofactors of corresponding elements of any other row (column) is ZERO. The value of determinant corresponding to a triangular determinant is equal to product of its principal diagonal elements. If in a determinant of odd order, a;; = —a;; and all diagonal elements are zero, then the value of determinant is zero. Kk. alae a oieltlel Multiplication of Two Determinants } by} #1\ (my) _ laity + byt. a, m+ b, mz by £>/ “Al Art, + botz az m,+ bz m2 (Row by column multiplication) Similarly, two determinants of order three are multiplied. Row by Row WO — =, "ee. Noy, Rak YM (ere a mH, es CaQjp bay Rev brkO> a) |b G 4.) [8 Gh (b) IfD=]a, by c2]}#0,thenD*?=]A, Bz Cz a3 b3 C3 Az Bz C3 where, Aj, B;, C; are cofactors of corresponding elements of D. $e Gears Cates ef caPocter determrrant) = (volo op Determrnant) Equations involving Three Variables } axtby+cez=d, _...(i) A2x+boy+c,z=dz (ii) a3x + bsy +¢3z = ds _.... (iii) Then, x =", y=" 7=3, (ig D+0) D D D du J Ue a by dq bh GY a, dy Where D =|a2 bz ¢2|/;D,=|dz bz C2|;D2=]a2 dz a3 bg C3 dz bz C3 a3 ds \coetBicient det [eal Cramer's Rule for Non-Homogeneous System of Equations axtbhytoz=d, sof ax + boy + Coz = dz Consistent —> Atleast one 3x + b3y + ¢3z = dg If D # 0 => Unique Solution. wee ue de Bar (stem se Consistent) If D = 0 & atleast one of D,, D, or D3 + 0 > No Solution ysern ® Pacansistent), If D = 0 & D, = D2 = D3 = 0 = Infinite Solution (SY 8 Songistent) D= d= 2,=B=0 =~ many sof) \ puovidec] tte 3 Een do Not yepresent Pal ~~ porate planer ind wrth a a8 X+yt2= / ty, * At Y+SS2 * KryreSVes a, D=d=BEvHFO Bot so0n a 2y3327 \ E Sa aa NN * + KAY} RzeR’ *, Ps 
