Evaluate $\begin{vmatrix} 4 & 2 & -1 \\ 2 & 3 & -1 \\ -1 & 1 & 3 \end{vmatrix}$

To evaluate the determinant of a $3 \times 3$ matrix, we use the expansion method along the first row.

1. Analyze the question

We are given the matrix:

We need to find the determinant, denoted as $|A|$.

2. Explain the concept

The determinant of a $3 \times 3$ matrix $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}$ is calculated as:

Remember the sign pattern for the first row is $(+, -, +)$.

3. Show the solution

Let's expand along the first row: $|A| = 4 \begin{vmatrix} 3 & -1 \\ 1 & 3 \end{vmatrix} - 2 \begin{vmatrix} 2 & -1 \\ -1 & 3 \end{vmatrix} + (-1) \begin{vmatrix} 2 & 3 \\ -1 & 1 \end{vmatrix}$

Step 1: Calculate the $2 \times 2$ determinants

• First part: $4[(3 \times 3) - (-1 \times 1)] = 4[9 - (-1)] = 4[10] = 40$
• Second part: $-2[(2 \times 3) - (-1 \times -1)] = -2[6 - 1] = -2[5] = -10$
• Third part: $-1[(2 \times 1) - (3 \times -1)] = -1[2 - (-3)] = -1[5] = -5$

Step 2: Combine the results $|A| = 40 - 10 - 5$ $|A| = 25$

4. Identify the answer

Comparing our result to the given options, the correct choice is A.

5. Exam Tip

When calculating determinants, be extremely careful with the signs of negative numbers inside the brackets (e.g., $-(-1)$ becomes $+1$). A common mistake is losing a negative sign during the multiplication of the $2 \times 2$ minors. Always write out the small brackets clearly to avoid these "silly" errors!