Let x be a non-zero real number. A determinant is chosen from the set of all determinants of order 2 with entries x or -x only. The probability that the value of the determinant is non-zero is

Options

A. `3/16`
B. `1/4`
C. `1/2`
D. `1/8`

Answer

To solve the problem, we need to find the probability that the value of a determinant of order 2, with entries being either \( x \) or \( -x \), is non-zero.

### Step-by-Step Solution:

1. **Understanding the Determinant of a 2x2 Matrix**:
The determinant of a 2x2 matrix is given by:
\[ \text{det} \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc \]
Here, \( a, b, c, d \) can take values \( x \) or \( -x \).

2. **Total Possible Matrices**:
Since each entry can independently be \( x \) or \( -x \), and there are 4 entries in a 2x2 matrix, the total number of different matrices is:
\[ 2^4 = 16 \]
This is because each of the 4 positions can be filled in 2 ways (either \( x \) or \( -x \)).

3. **Finding the Condition for Zero Determinant**:
We need to find the cases where the determinant is zero, i.e., when:
\[ ad = bc \]
We will analyze the combinations of \( a, b, c, d \) to find when this equality holds.

4. **Case Analysis**:
- **Case 1**: \( a = x \) and \( d = x \)
Then \( bc \) must also equal \( x^2 \). This can happen if:
- \( b = x \) and \( c = x \) (1 way)
- \( b = -x \) and \( c = -x \) (1 way)
- **Case 2**: \( a = -x \) and \( d = -x \)
Similar to Case 1:
- \( b = x \) and \( c = x \) (1 way)
- \( b = -x \) and \( c = -x \) (1 way)
- **Case 3**: \( a = x \) and \( d = -x \)
Then \( bc \) must equal \( -x^2 \). This can happen if:
- \( b = x \) and \( c = -x \) (1 way)
- \( b = -x \) and \( c = x \) (1 way)
- **Case 4**: \( a = -x \) and \( d = x \)
Similar to Case 3:
- \( b = x \) and \( c = -x \) (1 way)
- \( b = -x \) and \( c = x \) (1 way)

5. **Counting Favorable Cases**:
From the analysis above, we have:
- From Case 1: 2 ways
- From Case 2: 2 ways
- From Case 3: 2 ways
- From Case 4: 2 ways
Thus, the total number of cases where the determinant is zero is:
\[ 2 + 2 + 2 + 2 = 8 \]

6. **Calculating Probability of Zero Determinant**:
The probability that the determinant is zero is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P(\text{det} = 0) = \frac{\text{Number of cases where det = 0}}{\text{Total number of cases}} = \frac{8}{16} = \frac{1}{2} \]

7. **Finding the Probability of Non-Zero Determinant**:
The probability that the determinant is non-zero is:
\[ P(\text{det} \neq 0) = 1 - P(\text{det} = 0) = 1 - \frac{1}{2} = \frac{1}{2} \]

### Final Answer:
The probability that the value of the determinant is non-zero is \( \frac{1}{2} \).