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The problem asks to evaluate the determinant of the given 3x3 matrix \( A \) by selecting an optimal row or column to minimize the computations. The matrix is:

\[
A = \begin{bmatrix} 2 & 0 & 1 \\ -5 & 1 & 0 \\ 3 & -1 & 1 \end{bmatrix}
\]

### Step 1: Choose a row or column
A good choice is the second column because it contains two zeros, which will reduce the number of terms in the expansion.

\[
A = \begin{bmatrix} 2 & \mathbf{0} & 1 \\ -5 & \mathbf{1} & 0 \\ 3 & \mathbf{-1} & 1 \end{bmatrix}
\]

We can expand along the second column. The cofactor expansion formula for the determinant of a matrix is:

\[
\text{det}(A) = a_{1,j} \cdot C_{1,j} + a_{2,j} \cdot C_{2,j} + a_{3,j} \cdot C_{3,j}
\]

Where \( a_{i,j} \) is the element of the matrix at row \( i \), column \( j \), and \( C_{i,j} \) is the cofactor of that element.

### Step 2: Compute the determinant using cofactor expansion
The elements of the second column are \( 0, 1, -1 \). So the expansion becomes:

\[
\text{det}(A) = 0 \cdot C_{1,2} + 1 \cdot C_{2,2} + (-1) \cdot C_{3,2}
\]

Now we need to compute the cofactors \( C_{2,2} \) and \( C_{3,2} \).

- \( C_{2,2} \): This is the determinant of the 2x2 submatrix obtained by removing the second row and second column:

\[
C_{2,2} = \text{det} \begin{bmatrix} 2 & 1 \\ 3 & 1 \end{bmatrix} = (2)(1) - (1)(3) = 2 - 3 = -1
\]

- \( C_{3,2} \): This is the determinant of the 2x2 submatrix obtained by removing the third row and second column:

\[
C_{3,2} = \text{det} \begin{bmatrix} 2 & 1 \\ -5 & 0 \end{bmatrix} = (2)(0) - (1)(-5) = 0 + 5 = 5
\]

### Step 3: Substitute back into the expansion
Substitute the values of the cofactors back into the expansion:

\[
\text{det}(A) = 0 \cdot C_{1,2} + 1 \cdot (-1) + (-1) \cdot (5)
\]

\[
\text{det}(A) = -1 - 5 = -6
\]

### Final Answer:
The determinant of the matrix is \( \boxed{-6} \).

---

Would you like further clarification or details on any part of this process? Here are a few related questions to deepen your understanding:

1. How would you calculate the determinant if you expanded along a different row or column?
2. Why is choosing a row or column with zeros beneficial for determinant calculation?
3. What are the properties of cofactors that simplify determinant expansion?
4. How does the sign convention work when expanding the determinant using cofactors?
5. Can you explain the geometric interpretation of the determinant of a 3x3 matrix?

**Tip:** When calculating determinants, always try to pick a row or column with zeros to simplify the computation.

Evaluate the determinant of the matrix A = [[2, 0, 1], [-5, 1, 0], [3, -1, 1]] by carefully choosing a row or column to minimize computations.

Learn how to evaluate the determinant of the matrix A = [[2, 0, 1], [-5, 1, 0], [3, -1, 1]] by expanding along a selected column. This step-by-step solution simplifies the process using the cofactor expansion method, ideal for students at the college or advanced high school level.

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