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If P= [ array lll 1 & & 3 1 & 3 & 3 2 & 4 & 4 array ] is the adjoint of a 3 3 matrix A and |A|=4 then is equal to

Options

  1. A4
  2. B0
  3. C5
  4. D11

Correct answer

D. 11

Step-by-step solution

The property of the adjoint of a matrix states that |adj(A)| = |A|^ n-1 , where n is the order of the matrix A . Given n = 3 and |A| = 4 , we have |adj(A)| = |A|³⁻¹ = |A|^2 = 4^2 = 16 . The matrix P is given as P = adj(A) = bmatrix 1 & & 3 1 & 3 & 3 2 & 4 & 4 bmatrix . Calculating the determinant of P : |P| = 1(3 4 - 3 4) - (1 4 - 3 2) + 3(1 4 - 3 2) |P| = 1(12 - 12) - (4 - 6) + 3(4 - 6) |P| = 0 - (-2) + 3(-2) |P| = 2 - 6 . Equating |P| to 16 : 2 - 6 = 16 2 = 22 = 11 . Answer: 11

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