Example 18 - Show that A3 – 23A – 40I = O if A = [1 2 3 - Examples

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Example 18 If A = [ 8(1&2&3@3& 2&1@4&2&1)] then show that A3 23A 40I = O Finding A2 A2 = AA = [ 8(1&2&3@3& 2&1@4&2&1)] [ 8(1&2&3@3& 2&1@4&2&1)] = [ 8(1(1)+2(3)+3(4)&1(2)+2( 2)+3(2)&1(3)+2(1)+3(1)@3(1)+( 2)(3)+1(4)&3(2)+( 2)( 2)+1(2)&3(3)+( 2)(1)+1(1)@4(1)+2(3)+1(4)&4(2)+2( 2)+1(2)&4(3)+(2)(1)+1(1))] = [ 8(1+6+12&2 4+6&3+2+3@3 6+4&6+4+2&9 2+1@4+6+4&8 4+2&12+2+1)] = [ 8(19&4&8@1&12&8@14&6&15)] Finding A3 A3 =A2 A = [ 8(19&4&8@1&12&8@14&6&15)] [ 8(1&2&3@3& 2&1@4&2&1)] = [ 8(19(1)+4(3)+8(4)&19(2)+4( 2)+8(2)&19(3)+4(1)+8(1)@1(1)+12(3)+8(4)&1(2)+12( 2)+8(2)&1(3)+12(1)+8(1)@14(1)+6(3)+15(4)&14(2)+6( 2)+15(2)&14(3)+6(1)+15 (1))] = [ 8(19+12+32&38 8+16&57+4+8@1+36+32&2 24+16&3+12+8@14+18+60&28 12+30&42+6+15)] = [ 8(63&46&69@69& 6&23@92&46&63)] Calculating A3 23A 40I = [ 8(63&46&69@69& 6&23@92&46&63)] 23 [ 8(1&2&3@3& 2&1@4&2&1)] 40 [ 8(1&0&0@0&1&0@0&0&1)] = [ 8(63&46&69@69& 6&23@92&46&63)] [ 8(23 1&23 2&23 3@23 3&23 ( 2)&23 1@23 4&23 (2)&23 1)] [ 8(1 40&0 40&0 40@0 40&1 40&0 40@0 40&0 40&1 40)] = [ 8(63&46&69@69& 6&23@92&46&63)] 7 [ 8(23&46&69@69& 46&23@92&46&23)] 7 [ 8(40&0&0@0&40&0@0&0&40)] = [ 8(63 23 40&46 46+0&69 69+0@69 69+0& 6+46 40&23 23+0@92 92+0&46 46+0&63 23 40)] = [ 8(0&0&0@0&0&0@0&0&0)] = O 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.