Question

The largest value of the third determinant whose elements are equal to 1 or 0 is

• A. \(0\)
• B. \(2\)
• C. \(4\)
• D. \(6\)

Hint:

For a \(3\times3\) determinant with entries only \(0\) and \(1\), the maximum possible determinant value is \(2\).

Answer 1

The Correct Option is B

Concept:
A third-order determinant means a determinant of order: \[ 3\times3 \] The elements of the determinant are only: \[ 0 \text{ or } 1 \] We need the maximum possible determinant value.

Step 1: Understand determinant value.
A determinant represents a signed volume scaling factor. For a \(3\times3\) matrix with entries only \(0\) and \(1\), the determinant cannot become very large. The maximum possible value for such a determinant is a known result: \[ 2 \]

Step 2: Example giving determinant 2.
Consider the matrix: \[ \begin{vmatrix} 1 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 1 \end{vmatrix} \] Calculate: \[ =1(0\cdot1-1\cdot1)-1(1\cdot1-0\cdot1)+0 \] \[ =1( -1)-1(1) \] \[ =-1-1=-2 \] Its absolute value is: \[ 2 \] So determinant value can reach magnitude \(2\).

Step 3: Largest positive value.
By suitable row or column interchange, the determinant can be made: \[ +2 \] Therefore, the largest value is: \[ 2 \]

Step 4: Check the options.
Option (A) \(0\) is too small.
Option (B) \(2\) is correct.
Option (C) \(4\) is not possible for a \(3\times3\) determinant with only 0 and 1 entries.
Option (D) \(6\) is also not possible. Hence, the correct answer is: \[ \boxed{(B)\ 2} \]