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Let A = pmatrix 2 & 1 -3 & x pmatrix . Let , R be such that A^2 + A = 14 I , where I is the 2 2 identity matrix. If + = 6 , then the value of x is equal to

Options

  1. A1
  2. B-2
  3. C2
  4. D3

Correct answer

C. 2

Step-by-step solution

The characteristic equation of a 2 2 matrix A is given by A^2 - ( tr (A))A + ( (A))I = O . For the given matrix A , we have: tr (A) = 2 + x (A) = 2x - (-3) = 2x + 3 So, by the Cayley-Hamilton theorem: A^2 - (2+x)A + (2x+3)I = O A^2 - (2+x)A = -(2x+3)I Multiplying both sides by a constant k , we get: k A^2 - k(2+x)A = -k(2x+3)I Comparing this with the given equation A^2 + A = 14 I , we obtain: = k = -k(2+x) -k(2x+3) = 14 We are given that + = 6 . Substituting the expressions for and : k - k(2+x) = 6 k(1 - 2 - x) = 6

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