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Let A = bmatrix 1 & 0 & 0 3 & 1 & 0 9 & 3 & 1 bmatrix and B = [b_ ij ] , 1 i, j 3 . If B = A⁹⁹ - I , then the value of b₃₁ - b₂₁ b₃₂ is :

Options

  1. A99
  2. B199
  3. C149
  4. D159

Correct answer

C. 149

Step-by-step solution

Let A = I + C , where I is the identity matrix and C = bmatrix 0 & 0 & 0 3 & 0 & 0 9 & 3 & 0 bmatrix . Calculating the powers of C , we get: C^2 = bmatrix 0 & 0 & 0 3 & 0 & 0 9 & 3 & 0 bmatrix bmatrix 0 & 0 & 0 3 & 0 & 0 9 & 3 & 0 bmatrix = bmatrix 0 & 0 & 0 0 & 0 & 0 9 & 0 & 0 bmatrix C^3 = bmatrix 0 & 0 & 0 0 & 0 & 0 9 & 0 & 0 bmatrix bmatrix 0 & 0 & 0 3 & 0 & 0 9 & 3 & 0 bmatrix = bmatrix 0 & 0 & 0 0 & 0 & 0 0 & 0 & 0 bmatrix Since C^3 = 0 , all higher powers of C are also zero matrices. Using the binomial expan

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