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Ex 4.5, 14 - Find numbers a and b such that A2 + aA + bI = O

Ex 4.5, 14 - Chapter 4 Class 12 Determinants - Part 2
Ex 4.5, 14 - Chapter 4 Class 12 Determinants - Part 3 Ex 4.5, 14 - Chapter 4 Class 12 Determinants - Part 4

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Transcript

Ex 4.5, 14 For the matrix A = [■8(3&[email protected]&1)] , find the numbers a and b such that A2 + aA + bI = O. Finding A2 A2 = A.A = [■8(3&[email protected]&1)] [■8(3&[email protected]&1)] = [■8(3(3)+2(1)&3(2)+2(1)@1(3)+1(1)&1(2)+1(1))] = [■8(9+2&[email protected]+1&2+1)] = [■8(11&[email protected]&3)] Now, A2 + aA + bI = O Putting values [■8(11&[email protected]&3)] + a [■8(3&[email protected]&1)] + b [■8(1&[email protected]&1)] = O [■8(11&[email protected]&3)] + [■8(3a&[email protected]&a)] + [■8(b&[email protected]&b)] = O [■8(11+3a+b&[email protected]+a+0&3+a+b)] = O [■8(3a+b+11&[email protected]+a&a+b+3)] = [■8(0&[email protected]&0)] Since the matrices are equal, Comparing corresponding elements 3a + b + 11 = 0 2a + 8 = 0 4 + a = 0 a + b + 3 = 0 Solving (3) a + 4 = 0 a = –4 Putting value a in (1) 11 + 3 a + b = 0 11 + 3 (–4) + b = 0 11 – 12 +b = 0 –1 + b = 0 …(1) b = 1 Hence, a = −4, b = 1

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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.