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

Example 18 - Chapter 3 Class 12 Matrices - Part 2
Example 18 - Chapter 3 Class 12 Matrices - Part 3

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Transcript

Example 18 If A = [■8(1&2&[email protected]&−2&[email protected]&2&1)] then show that A3 – 23A – 40I = O Finding A2 A2 = AA = [■8(1&2&[email protected]&−2&[email protected]&2&1)] [■8(1&2&[email protected]&−2&[email protected]&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&[email protected]−6+4&6+4+2&9−[email protected]+6+4&8−4+2&12+2+1)] = [■8(19&4&[email protected]&12&[email protected]&6&15)] Finding A3 A3 = A2 A = [■8(19&4&[email protected]&12&[email protected]&6&15)] [■8(1&2&[email protected]&−2&[email protected]&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&[email protected]+36+32&2−24+16&[email protected]+18+60&28−12+30&42+6+15)] = [■8(63&46&[email protected]&−6&[email protected]&46&63)] Calculating A3 – 23A – 40I = [■8(63&46&[email protected]&−6&[email protected]&46&63)] −23 [■8(1&2&[email protected]&−2&[email protected]&2&1)] −40 [■8(1&0&[email protected]&1&[email protected]&0&1)] = [■8(63&46&[email protected]&−6&[email protected]&46&63)] −[■8(23×1&23×2&23×[email protected]×3&23×(−2)&23×[email protected]×4&23×(2)&23×1)] − [■8(1×40&0×40&0×[email protected]×40&1×40&0×[email protected]×40&0×40&1×40)] = [■8(63&46&[email protected]&−6&[email protected]&46&63)] ⤶7− [■8(23&46&[email protected]&−46&[email protected]&46&23)] ⤶7− [■8(40&0&[email protected]&40&[email protected]&0&40)] = [■8(63−23−40&46−46+0&69−[email protected]−69+0&−6+46−40&23−[email protected]−92+0&46−46+0&63−23−40)] = [■8(0&0&[email protected]&0&[email protected]&0&0)] = O Hence proved.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.