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Question Number 57619 by tanmay.chaudhury50@gmail.com last updated on 08/Apr/19

Answered by math1967 last updated on 09/Apr/19

determinant (((q+r),(r+p),(p+q)),((y+z),(z+x),(x+y))) determinant (((b+c−c−a−a−b),(c+a),(a+b)),((q+r−r−p−p−q),(r+p),(p+q)),((y+z−z−x−x−y),(z+x),(x+y)))C_1 −C_2 −C_3 −2 determinant ((a,(c+a),(a+b)),(p,(r+p),(p+q)),(x,(z+x),(x+y))) =−2 determinant ((a,(c+a−a),(a+b−a)),(p,(r+p−p),(p+q−p)),(x,(z+x−x),(x+y−x)))C_2 −C_1 ,C_3 −C_1 =−2 determinant ((a,c,b),(p,r,q),(x,z,(y )))=2 determinant ((a,b,c),(p,q,r),(x,y,z))C_2 ⇔C_3

|q+rr+pp+qy+zz+xx+y||b+c−c−a−a−bc+aa+bq+r−r−p−p−qr+pp+qy+z−z−x−x−yz+xx+y|C1−C2−C3−2|ac+aa+bpr+pp+qxz+xx+y|=−2|ac+a−aa+b−apr+p−pp+q−pxz+x−xx+y−x|C2−C1,C3−C1=−2|acbprqxzy|=2|abcpqrxyz|C2⇔C3

Commented by tanmay.chaudhury50@gmail.com last updated on 09/Apr/19

thank you sir

thankyousir

Commented by math1967 last updated on 09/Apr/19

You are welcome sir

Youarewelcomesir

Answered by tanmay.chaudhury50@gmail.com last updated on 09/Apr/19

another way ∣b+c c+a a+b ∣ ∣q+r r+p p+q∣ ∣y+z z+x x+y∣ =∣b c a∣ +∣b c b∣ +∣b a a∣+∣b a b∣ ∣q r p∣ ∣q r q∣ ∣q p p∣ ∣q p q∣ + ∣ y z x∣ ∣y z y∣ ∣y x x∣ ∣y x y∣ ∣c c a∣+∣c c b ∣+∣c a a∣+∣c a b∣ ∣r r p∣ ∣r r q∣ ∣r p p∣ ∣r p q∣ ∣z z x∣ ∣z z y∣ ∣z x x∣ ∣z x y∣ △=△_1 +△_2 +△_3 +△_4 +△_5 +△_6 +△_7 +△_8 now △_2 =△_3 =△_4 =△_5 =△_6 =△_7 =0 reason two identical collumn so △=△_1 +△_8 now inter change of collumn keep value of determinant unchanged So △=2∣a b c∣ ∣p q r∣ ∣x y z∣ proved

anotherway∣b+cc+aa+b∣∣q+rr+pp+q∣∣y+zz+xx+y∣=∣bca∣+∣bcb∣+∣baa∣+∣bab∣∣qrp∣∣qrq∣∣qpp∣∣qpq∣+∣yzx∣∣yzy∣∣yxx∣∣yxy∣∣cca∣+∣ccb∣+∣caa∣+∣cab∣∣rrp∣∣rrq∣∣rpp∣∣rpq∣∣zzx∣∣zzy∣∣zxx∣∣zxy∣△=△1+△2+△3+△4+△5+△6+△7+△8now△2=△3=△4=△5=△6=△7=0reasontwoidenticalcollumnso△=△1+△8nowinterchangeofcollumnkeepvalueofdeterminantunchangedSo△=2∣abc∣∣pqr∣∣xyz∣proved

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