Given matrix \( A = \left[\begin{matrix} x & 1 & 3 \\ y & 2 & 6 \\ 3 & 5 & 7 \end{matrix}\right] \), the ordered pair \( (x, y) \) for which \( \text{det}(A) = 0 \) is
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- (1, 1)
- (1, 2)
- (2, 2)
- (2, 1)
The Correct Option is B
Solution and Explanation
We are given the matrix \( A = \left[\begin{matrix} x & 1 & 3 \\ y & 2 & 6 \\ 3 & 5 & 7 \end{matrix}\right] \), and we need to find the ordered pair \( (x, y) \) such that the determinant of \( A \) is zero.
Step 1: Calculate the determinant of \( A \). The determinant of \( A \) is: \[ \text{det}(A) = x \cdot \begin{vmatrix} 2 & 6 \\ 5 & 7 \end{vmatrix} - 1 \cdot \begin{vmatrix} y & 6 \\ 3 & 7 \end{vmatrix} + 3 \cdot \begin{vmatrix} y & 2 \\ 3 & 5 \end{vmatrix}. \] First, compute the 2x2 determinants: \[ \begin{vmatrix} 2 & 6 \\ 5 & 7 \end{vmatrix} = 2 \times 7 - 6 \times 5 = 14 - 30 = -16, \] \[ \begin{vmatrix} y & 6 \\ 3 & 7 \end{vmatrix} = y \times 7 - 6 \times 3 = 7y - 18, \] \[ \begin{vmatrix} y & 2 \\ 3 & 5 \end{vmatrix} = y \times 5 - 2 \times 3 = 5y - 6. \] Step 2: Substitute these values back. Substitute the results into the determinant equation: \[ \text{det}(A) = x \cdot (-16) - 1 \cdot (7y - 18) + 3 \cdot (5y - 6). \] Simplify the equation: \[ \text{det}(A) = -16x - 7y + 18 + 15y - 18 = -16x + 8y. \] For \( \text{det}(A) = 0 \), we get: \[ -16x + 8y = 0. \] Solving for \( y \), we get: \[ y = 2x. \] Step 3: Identify the solution. The ordered pair \( (x, y) \) that satisfies this equation is \( (2, 1) \), as \( x = 2 \) and \( y = 1 \) satisfy the equation.
Final Answer: (2, 1)
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