What is the largest value of a third order determinant whose elements are 0 or 1 ?

Options

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

Answer

To find the largest value of a third order determinant whose elements are either 0 or 1, we can follow these steps:

### Step 1: Define the Determinant
Let \( D \) be a 3x3 determinant represented as follows:

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

where each \( a_{ij} \) (element of the determinant) can either be 0 or 1.

### Step 2: Consider Possible Configurations
To maximize the determinant, we should try different configurations of 0s and 1s. A common approach is to set the diagonal elements to 1 and the off-diagonal elements to 0, or vice versa.

### Step 3: Calculate the Determinant
Let's consider the following configuration:

\[ D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{vmatrix} \]

Now, we can calculate the determinant using the formula for a 3x3 determinant:

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

Plugging in the values:

\[ D = 1(1 \cdot 1 - 0 \cdot 0) - 1(1 \cdot 1 - 0 \cdot 1) + 1(1 \cdot 0 - 1 \cdot 1) \]

Calculating each term:

1. First term: \( 1(1 - 0) = 1 \)
2. Second term: \( -1(1 - 0) = -1 \)
3. Third term: \( 1(0 - 1) = -1 \)

Combining these:

\[ D = 1 - 1 - 1 = -1 \]

### Step 4: Try Another Configuration
Now, let's try another configuration:

\[ D = \begin{vmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{vmatrix} \]

Calculating this determinant:

\[ D = 1(1 \cdot 1 - 0 \cdot 0) - 0(0 \cdot 1 - 0 \cdot 0) + 0(0 \cdot 0 - 1 \cdot 0) = 1 \]

### Step 5: Explore More Configurations
We can also try:

\[ D = \begin{vmatrix} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{vmatrix} \]

Calculating this determinant:

\[ D = 1(0 \cdot 1 - 1 \cdot 1) - 1(1 \cdot 1 - 0 \cdot 1) + 0(1 \cdot 1 - 0 \cdot 1) \]

Calculating:

1. First term: \( 1(0 - 1) = -1 \)
2. Second term: \( -1(1 - 0) = -1 \)
3. Third term: \( 0 \)

Combining these:

\[ D = -1 - 1 + 0 = -2 \]

### Conclusion
After testing various configurations, the maximum value of the determinant we found is 1. However, we can also find configurations that yield a determinant of 2.

The largest value of a third order determinant whose elements are 0 or 1 is:

\[ \boxed{2} \]