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Let A = bmatrix 3 & -4 1 & -1 bmatrix and B = bmatrix 6 & -13 5 & -10 bmatrix be two matrices. If the variables x and y satisfy the matrix equation ((A⁻¹)^2 + B) bmatrix x y bmatrix = bmatrix 0 0 bmatrix , then the ordered pair (x, y) =

Options

  1. A(3, 5)
  2. B(10, 7)
  3. C(4, 6)
  4. D(5, 3)

Correct answer

D. (5, 3)

Step-by-step solution

Given A = bmatrix 3 & -4 1 & -1 bmatrix Determinant of A is |A| = (3)(-1) - (-4)(1) = 1 The inverse of A is A⁻¹ = 1 |A| bmatrix -1 & 4 -1 & 3 bmatrix = bmatrix -1 & 4 -1 & 3 bmatrix Now, (A⁻¹)^2 = bmatrix -1 & 4 -1 & 3 bmatrix bmatrix -1 & 4 -1 & 3 bmatrix = bmatrix 1 - 4 & -4 + 12 1 - 3 & -4 + 9 bmatrix = bmatrix -3 & 8 -2 & 5 bmatrix Adding matrix B : (A⁻¹)^2 + B = bmatrix -3 & 8 -2 & 5 bmatrix + bmatrix 6 & -13 5 & -10 bmatrix = bmatrix 3 & -5 3 & -5 bmatrix The given matrix equation is ((A⁻¹)^2 + B) bmatrix x y

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