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Ex 3.2, 16 - Prove A3 - 6A2 + 7A + 2I = O, given A = [1 0

Ex 3.2, 16 - Chapter 3 Class 12 Matrices - Part 2
Ex 3.2, 16 - Chapter 3 Class 12 Matrices - Part 3 Ex 3.2, 16 - Chapter 3 Class 12 Matrices - Part 4

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Ex 3.2, 16 If A = [■8(1&0&[email protected]&2&[email protected]&0&3)] , prove that A3 – 6A2 + 7A + 2I = O Finding A2 A2 = A × A = [■8(1&0&[email protected]&2&[email protected]&0&3)] [■8(1&0&[email protected]&2&[email protected]&0&3)] = [■8(1(1)+0 (0)+2(2)&1(0)+0(2)+2(0)&1(2)+0(1)+2(3)@0(1)+2(0)+1(2)&0(0)+2(2)+1(0)&0(2)+2(1)+1(3)@2(1)+0(0)+3(2)&2(0)+0(2)+3(0)&2(2)+0(1)+3(3))] = [■8(1+0+4&0+0+0&[email protected]+0+2&0+4+0&[email protected]+0+6&0+0+0&4+0+9)] = [■8(5&0&[email protected]&4&[email protected]&0&13)] Finding A3 A3 = A2. A = [■8(5&0&[email protected]&4&[email protected]&0&13)] [■8(1&0&[email protected]&2&[email protected]&0&3)] = [■8(5(1)+0 (0)+8(2)&5(0)+0(2)+8(0)&5(2)+0(1)+8(3)@2(1)+4(0)+5(2)&2(0)+4(2)+5(0)&2(2)+4(1)+5(3)@8(1)+0(0)+13(2)&8(0)+0(2)+13(0)&8(2)+0(1)+13(3))] = [■8(5+0+16&0+0+0&[email protected]+0+10&0+8+0&[email protected]+0+26&0+0+0&16+0+39)] = [■8(21&0&[email protected]&8&[email protected]&0&55)] Now calculating A3 - 6A2 +7A + 2I Putting values = [■8(21&0&[email protected]&8&[email protected]&0&55)] – 6 [■8(5&0&[email protected]&4&[email protected]&0&13)] + 7 [■8(1&0&[email protected]&2&[email protected]&0&3)] + 2 [■8(1&0&[email protected]&1&[email protected]&0&1)] = [■8(21&0&[email protected]&8&[email protected]&0&55)] – [■8(6(5)&0(5)&8(6)@2(6)&4(6)&5(6)@8(6)&0(6)&13(6))] + [■8(1(7)&0(7)&2(7)@0(7)&2(7)&1(7)@2(7)&0(7)&3(7))] + [■8(2(1)&2(0)&2(0)@2(0)&1(2)&0(2)@2(0)&0(2)&1(2))] = [■8(21−30+7+2&0−0+0+0&34−[email protected]−12+0+0&8−24+14+2&23−[email protected]−48+14+0&0+0+0+0&55−78+21+2)] = [■8(−30+30&0&−[email protected]−12&24−24&30−[email protected]−48&0&78−78)] = [■8(0&0&[email protected]&0&[email protected]&0&0)] = O Thus, A3 – 6A2 + 7A + 2I = O Hence proved

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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.