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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise
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Ex 4.5, 4 Verify A (adj A) = (adj A) A = | |I for A = [ 8(1& 1&2@3&0& 2@1&0&3)] Calculating | | |A| = | 8(1& 1&2@3&0& 2@1&0&3)| = 1 | 8(0& 2@0&3)| ( 1) | 8(3& 2@1&3)| +2 | 8(3&0@1&0)| = 1 (0 0) + 1 (9 + 2) +2 (0 0) = 11 Calculating adj A adj A = [ 8(A_11&A_21&A_31@A_12&A_22&A_32@A_13&A_23&A_33 )] A = [ 8(1& 1&2@3&0& 2@1&0&3)] M11 = | 8(0& 2@0&3)| = 0(3) 0( 2) = 0 M12 = | 8(3& 2@1&3)| = 3(3) 1( 2) = 11 M13 = | 8(3&0@1&0)| = 3(0) 0(1) = 0 M21 = | 8( 1&2@0&3)| = 1(3) 0(2) = 3 M22 = | 8(1&2@1&3)| = 1(3) 1(2) = 1 M23 = | 8(1& 1@1&0)| = 1(0) 1( 1) = 1 M31 = | 8("-" 1&2@0&"-" 2)| = -1( 2) 0(2) = 2 M32 = | 8(1&2@3& 2)| = 1( 2) 3(2) = 8 M33 = | 8(1& 1@3&0)| = 1(0) 3( 1) = 3 A11 = ( 1)1 + 1 M11 = ( 1)2 0 = 0 A12 = ( 1)1+2 M12 = ( 1)3 (11) = 11 A13 = ( 1)1+3 M13 = ( 1)4 0 = 0 A21 = ( 1)2+1 M21 = ( 1)3 ( 3) = 3 A22 = ( 1)2+2 M22 = ( 1)4 . 1 = 1 A23 = ( 1)2+3 M23 = ( 1)5 ( 1) = 1 A31 = ( 1)3+1 M31 = ( 1)4 (2) = 2 A32 = ( 1)3+2 M32 = ( 1)5 ( 8) = 8 A33 = ( 1)3+3 M33 = ( 1)6 ( 3) = 3 Thus adj (A) = [ 8(A11&A21&A31@A12&A22&A32@A33&A23&A33)] = [ 8(0&3&2@ 11&1&8@0& 1&3)] Calculating A (adj A) [ 8(1& 1&2@3&0& 2@1&0&3)] [ 8(0&3&2@ 11&1&8@0& 1&3)] = [ 8(1(0) 1( 7 11)+2(0)&1(3) 1(1)+2( 1)&1(2) 1(8)+2(3)@3(0)+0( 7 11)+( 2)(0)&3(3)+0(1)+( 2)( 1)&3(2)+0(8)+( 2)(3)@1(0)+0( 7 11)+3(0)&1(3)+0(1)+3( 1)&1(2)+0(8)+3(3))] = [ 8(0+11+0&3 1 2&2 8+6@0 0 0&9+0+2&6+0 6@0 0+0&3+0 3&2+0+9)] = [ 8(11&0&0@0&11&0@0&0&11)] = 11 [ 8(1&0&0@0&1&0@0&0&1)] = 11I Calculating (adj A)A [ 8(0&3&2@ 11&1&8@0& 1&3)] [ 8(1& 1&2@3&0& 2@1&0&3)] = [ 8(0(1)+3(3)+2(1)&0( 1)+3(0)+2(0)&0(2)+3( 2)+2(3)@ 11(1)+1(3)+8(1)& 11( 1)+1(0)+8(0)& 11(2)+1( 2)+8(3)@0(1)+( 1)(3)+3(1)&0( 1)+( 1)(0)+3(0)&0(2)+( 1)( 2)+3(3))] = [ 8(0+9+2& 0+0+0&0 6+6@ 11+3+8&11+0+0& 22 2+24@0 3+3& 0 0+0& 0+2+9)] = [ 8(11&0&0@0&11&0@0&0&11)] = 11 [ 8(1&0&0@0&1&0@0&0&1)] = 11I Calculating |A| I |A|I = 11I Thus, A (adj(A)) = (adj A) A = |A| I = 11I A (adj(A)) = (adj A) A = |A| I Hence Proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.