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

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  1. Chapter 4 Class 12 Determinants
  2. Serial order wise
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Ex 4.5, 14 For the matrix A = 3﷮2﷮1﷮1﷯﷯ , find the numbers a and b such that A2 + aA + bI = O. Finding A2 A2 = A.A = 3﷮2﷮1﷮1﷯﷯ 3﷮2﷮1﷮1﷯﷯ = 3(3)+2(1)﷮3(2)+2(1)﷮1(3)+1(1)﷮1(2)+1(1)﷯﷯ = 9+2﷮6+2﷮3+1﷮2+1﷯﷯ = 11﷮8﷮4﷮3﷯﷯ Now, A2 + aA + bI = O Putting values 11﷮8﷮4﷮3﷯﷯ + a 3﷮2﷮1﷮1﷯﷯ + b 1﷮0﷮0﷮1﷯﷯ = O 11﷮8﷮4﷮3﷯﷯ + 3a﷮2a﷮a﷮a﷯﷯ + b﷮0﷮0﷮b﷯﷯ = O 11+3a+b﷮8+2a+0﷮4+a+0﷮3+a+b﷯﷯ = O 3a+b+11﷮2a+8﷮4+a﷮a+b+3﷯﷯ = O 3a+b+11﷮2a+8﷮4+a﷮a+b+3﷯﷯ = 0﷮0﷮0﷮0﷯﷯ Since the matrices are equal, Comparing corresponding elements 3a + b + 11 = 1 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 b = 1 Hence, a = −4, b = 1

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