Minimum value of a second order determinant whose each is either 1 or 2 is equal to

Options

A. 0
B. -1
C. -2
D. -3

Answer

To find the minimum value of a second order determinant where each element is either 1 or 2, we can follow these steps:

1. **Define the Determinant**:
A second order determinant can be represented as:
\[ \text{Det}(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \]
where \(a\), \(b\), \(c\), and \(d\) can each take values of either 1 or 2.

2. **List Possible Combinations**:
Since each element can be either 1 or 2, we can list all possible combinations of \(a\), \(b\), \(c\), and \(d\). The combinations are:
- (1, 1, 1, 1)
- (1, 1, 1, 2)
- (1, 1, 2, 1)
- (1, 1, 2, 2)
- (1, 2, 1, 1)
- (1, 2, 1, 2)
- (1, 2, 2, 1)
- (1, 2, 2, 2)
- (2, 1, 1, 1)
- (2, 1, 1, 2)
- (2, 1, 2, 1)
- (2, 1, 2, 2)
- (2, 2, 1, 1)
- (2, 2, 1, 2)
- (2, 2, 2, 1)
- (2, 2, 2, 2)

3. **Calculate the Determinant for Each Combination**:
We will calculate \(ad - bc\) for each combination:
- For (1, 1, 1, 1): \(1 \cdot 1 - 1 \cdot 1 = 0\)
- For (1, 1, 1, 2): \(1 \cdot 2 - 1 \cdot 1 = 1\)
- For (1, 1, 2, 1): \(1 \cdot 1 - 1 \cdot 2 = -1\)
- For (1, 1, 2, 2): \(1 \cdot 2 - 1 \cdot 2 = 0\)
- For (1, 2, 1, 1): \(1 \cdot 1 - 2 \cdot 1 = -1\)
- For (1, 2, 1, 2): \(1 \cdot 2 - 2 \cdot 1 = 0\)
- For (1, 2, 2, 1): \(1 \cdot 1 - 2 \cdot 2 = -3\)
- For (1, 2, 2, 2): \(1 \cdot 2 - 2 \cdot 2 = -2\)
- For (2, 1, 1, 1): \(2 \cdot 1 - 1 \cdot 1 = 1\)
- For (2, 1, 1, 2): \(2 \cdot 2 - 1 \cdot 1 = 3\)
- For (2, 1, 2, 1): \(2 \cdot 1 - 1 \cdot 2 = 0\)
- For (2, 1, 2, 2): \(2 \cdot 2 - 1 \cdot 2 = 2\)
- For (2, 2, 1, 1): \(2 \cdot 1 - 2 \cdot 2 = -2\)
- For (2, 2, 1, 2): \(2 \cdot 2 - 2 \cdot 1 = 2\)
- For (2, 2, 2, 1): \(2 \cdot 1 - 2 \cdot 2 = -2\)
- For (2, 2, 2, 2): \(2 \cdot 2 - 2 \cdot 2 = 0\)

4. **Find the Minimum Value**:
After calculating the determinants for all combinations, we find that the minimum value is \(-3\) from the combination (1, 2, 2, 1).

Thus, the minimum value of the second order determinant whose each element is either 1 or 2 is:
\[ \boxed{-3} \]