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Let M(x) = bmatrix 1-x & x -x & 1+x bmatrix be a matrix defined for all real numbers x . Which of the following relations holds true for all real x and y ?

Options

  1. AM(x)M(y) = M(xy) and M(x)⁻¹ = M ( 1 x )
  2. BM(x)M(y) = M(x+y) and M(x)⁻¹ = M(-x)
  3. CM(x)M(y) = M(x-y) and M(x)⁻¹ = M(-x)
  4. DM(x)M(y) = M(x+y) and M(x)⁻¹ = M ( 1 x )

Correct answer

B. M(x)M(y) = M(x+y) and M(x)⁻¹ = M(-x)

Step-by-step solution

Given M(x) = bmatrix 1-x & x -x & 1+x bmatrix and M(y) = bmatrix 1-y & y -y & 1+y bmatrix . Let us find the product M(x)M(y) using standard row-by-column matrix multiplication: M(x)M(y) = bmatrix 1-x & x -x & 1+x bmatrix bmatrix 1-y & y -y & 1+y bmatrix The elements of the resulting matrix are: Row 1, Column 1: (1-x)(1-y) + x(-y) = 1 - x - y + xy - xy = 1 - (x+y) Row 1, Column 2: (1-x)y + x(1+y) = y - xy + x + xy = x+y Row 2, Column 1: -x(1-y) + (1+x)(-y) = -x + xy - y - xy = -(x+y) Row 2, Column 2: -xy + (1+x)(1+y

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