Question

If the entries in a $3\times 3$ determinant are either $0$ or 1, then the greatest value of their determinats is:

• A. 1
• B. 2
• C. 3
• D. 9

Answer 1

Answer: (B)

2

Case I: If all the entries in one row are 0 , then the determinant is 0.
Case II: If all the entries in one row are 1 , then the maximum value of determinant is 1.
$\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 1& 0& 1\end{array}\right]$
Case II:If exactly one 1 is in one row, then the maximum value is 1.
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Case III:If exactly two 1 is in one row, then the maximum value is 2.
$\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 1& 0& 1\end{array}\right]$
Hence, the largest value of the determinant with entries either 0 or 1 is 2.