Question

Let \[ f(x)= \begin{vmatrix} -1 & x & 3 \\ 0 & 1 & 2x \\ 1 & -1 & 1 \end{vmatrix}. \] Then the value of \(f(-1)\) is equal to:

• A. 6
• B. -4
• C. -2
• D. 2
• E. 0

Hint:

Always look for zeros in a column or row to expand the determinant. Expanding along the first column here was the fastest choice due to the zero in the middle.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The function \(f(x)\) is defined as a \(3 \times 3\) determinant. To find the value at a specific point, we substitute the value of \(x\) into the determinant and evaluate it.

Step 2: Key Formula or Approach:
Substitute \(x = -1\) into the matrix and expand along any row or column.

Step 3: Detailed Explanation:
Substituting \(x = -1\) into the determinant:
\[ f(-1) = \begin{vmatrix} -1 & -1 & 3 \\ 0 & 1 & 2(-1) \\ 1 & -1 & 1 \end{vmatrix} = \begin{vmatrix} -1 & -1 & 3 \\ 0 & 1 & -2 \\ 1 & -1 & 1 \end{vmatrix} \]
Expanding along the first column:
\[ f(-1) = -1 \cdot [(1)(1) - (-2)(-1)] - 0 \cdot [(-1)(1) - (3)(-1)] + 1 \cdot [(-1)(-2) - (3)(1)] \]
Simplify each term:
\[ f(-1) = -1 \cdot [1 - 2] - 0 + 1 \cdot [2 - 3] \]
\[ f(-1) = -1 \cdot [-1] + 1 \cdot [-1] \]
\[ f(-1) = 1 - 1 = 0 \]

Step 4: Final Answer:
The value of \(f(-1)\) is 0.