If a determinant of order `3xx3` is formed by using the numbers 1 or -1 then minimum value of determinant is :

Options

A. -2
B. -4
C. 0
D. -8

Explanation

To find the minimum value of a determinant of order 3 formed by using the numbers 1 or -1, we can follow these steps:

### Step 1: Define the Matrix
Let’s denote the 3x3 matrix formed by the numbers 1 and -1 as follows:

\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

where each \(a_{ij}\) can be either 1 or -1.

### Step 2: Calculate the Determinant
The determinant of a 3x3 matrix is given by the formula:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

### Step 3: Consider All Possible Combinations
Since each entry can be either 1 or -1, there are \(2^9 = 512\) possible combinations of entries in the matrix. However, we are interested in finding the minimum value of the determinant.

### Step 4: Analyze the Determinant
To minimize the determinant, we can observe that the determinant can take negative values. The determinant will be minimized when we have a configuration of 1s and -1s that leads to the most negative value.

### Step 5: Finding the Minimum Value
Through analysis or by testing various configurations, we can find that the minimum value of the determinant occurs when the matrix is structured in a certain way. For example, if we set:

\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & -1 \end{pmatrix} \]

Calculating the determinant of this matrix:

\[ \text{det}(A) = 1(1 \cdot (-1) - (-1) \cdot (-1)) - 1(1 \cdot (-1) - (-1) \cdot 1) + 1(1 \cdot (-1) - 1 \cdot 1) \]

This simplifies to:

\[ = 1(-1 - 1) - 1(-1 - (-1)) + 1(-1 - 1) = 1(-2) - 1(0) + 1(-2) = -2 - 2 = -4 \]

Thus, the minimum value of the determinant is \(-4\).

### Final Result
The minimum value of the determinant formed by using the numbers 1 and -1 is:

\[ \text{Minimum Value} = -4 \]