To solve the problem, we need to find the probability that a 2x2 determinant, with entries chosen from the set \(\{-1, 1\}\), has a value of zero.
### Step-by-Step Solution:
1. **Understanding the Determinant**:
The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is calculated as:
\[
\text{det} = ad - bc
\]
We want to find the cases where this determinant equals zero:
\[
ad - bc = 0 \implies ad = bc
\]
2. **Possible Values for Entries**:
Since \(a\), \(b\), \(c\), and \(d\) can each be either \(-1\) or \(1\), we can list the possible combinations for \(a\), \(b\), \(c\), and \(d\). Each entry has 2 choices, leading to a total of:
\[
2^4 = 16 \text{ total combinations}
\]
3. **Finding Favorable Cases**:
We need to find the combinations where \(ad = bc\). We can analyze the possible values of \(ad\) and \(bc\):
- If \(ad = 1\), then \(bc\) must also be \(1\).
- If \(ad = -1\), then \(bc\) must also be \(-1\).
Let's consider each case:
- **Case 1**: \(ad = 1\)
- This occurs when both \(a\) and \(d\) are either \(1\) or both are \(-1\). Thus, we have:
- \(a = 1, d = 1\) → \(bc\) must be \(1\) (i.e., \(b\) and \(c\) can be \((1, 1)\) or \((-1, -1)\)).
- \(a = -1, d = -1\) → \(bc\) must be \(1\) (i.e., \(b\) and \(c\) can be \((1, 1)\) or \((-1, -1)\)).
- This gives us 4 combinations:
1. \( (1, 1, 1, 1) \)
2. \( (1, 1, -1, -1) \)
3. \( (-1, -1, 1, 1) \)
4. \( (-1, -1, -1, -1) \)
- **Case 2**: \(ad = -1\)
- This occurs when one of \(a\) or \(d\) is \(1\) and the other is \(-1\). Thus, we have:
- \(a = 1, d = -1\) → \(bc\) must be \(-1\) (i.e., \(b\) and \(c\) can be \((1, -1)\) or \((-1, 1)\)).
- \(a = -1, d = 1\) → \(bc\) must be \(-1\) (i.e., \(b\) and \(c\) can be \((1, -1)\) or \((-1, 1)\)).
- This gives us 4 combinations:
5. \( (1, 1, -1, 1) \)
6. \( (1, -1, 1, -1) \)
7. \( (-1, 1, 1, -1) \)
8. \( (-1, -1, -1, 1) \)
In total, we have \(4 + 4 = 8\) favorable cases where the determinant is zero.
4. **Calculating the Probability**:
The probability \(P\) that the determinant is zero is given by the ratio of favorable cases to total cases:
\[
P = \frac{\text{Number of favorable cases}}{\text{Total cases}} = \frac{8}{16} = \frac{1}{2}
\]
### Final Answer:
The probability that the determinant has a zero value is \(\frac{1}{2}\).
