Maths · Determinants and matrices of order two and three
If \) for then is:
If \( \boldsymbol{A}=\left[\begin{array}{ccc}\mathbf{1} & \mathbf{1} & \mathbf{1} \\ \mathbf{1} & \mathbf{1}+\boldsymbol{x} & \mathbf{1} \\ \mathbf{1} & \mathbf{1} & \mathbf{1}+\boldsymbol{y}\end{array}\right] \) for \( \boldsymbol{x} \neq \) \( \mathbf{0}, \boldsymbol{y} \neq \mathbf{0}, \) then \( \boldsymbol{D} \) is:
- A. divisible by neither \( x \) nor \( y \)
- B. divisible by both \( x \) nor \( y \)
- C. divisible by \( x \) but not \( y \)
- D. divisible by \( y \) but not \( x \)
Step-by-step solution
The determinant of matrix A is computed by row operations: subtract row1 from row2 and row3 gives a triangular matrix with diagonal entries 1, x, y, so det = 1*x*y = xy. Since x≠0 and y≠0, the determinant is divisible by both x and y.
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