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If A = pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix , then what is A^2 - 4A equal to?

Options

  1. A-5I₃
  2. B-I₃
  3. CI₃
  4. D5I₃

Correct answer

D. 5I₃

Step-by-step solution

Given A = pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix First, calculating A^2 : A^2 = pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix A^2 = pmatrix 1+4+4 & 2+2+4 & 2+4+2 2+2+4 & 4+1+4 & 4+2+2 2+4+2 & 4+2+2 & 4+4+1 pmatrix = pmatrix 9 & 8 & 8 8 & 9 & 8 8 & 8 & 9 pmatrix Now, calculating 4A : 4A = 4 pmatrix 1 & 2 & 2 2 & 1 & 2 2 & 2 & 1 pmatrix = pmatrix 4 & 8 & 8 8 & 4 & 8 8 & 8 & 4 pmatrix Subtracting 4A from A^2 : A^2 - 4A = pmatrix 9 & 8 & 8 8 & 9 & 8 8 & 8 & 9 pmatrix

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