If the entries in a 3 x 3 determinant are either 0 or 1 , then the greatest value of this determinant is :a)2b)1c)0d)none of these.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

Step 1: Let us write the general expression for a 3 x 3 determinant with entries 0 or 1.
$\begin{vmatrix}
a & b & c \
d & e & f \
g & h & i
\end{vmatrix}$
where a, b, c, d, e, f, g, h, i $\in$ {0, 1}

Step 2: We need to find the maximum value of this determinant. Let us consider the cases where the determinant will be 2.

Case 1: When the diagonal elements are all 1
$\begin{vmatrix}
1 & b & c \
d & 1 & f \
g & h & 1
\end{vmatrix}$
= 1 $\cdot$ 1 $\cdot$ 1 + d $\cdot$ b $\cdot$ h + g $\cdot$ f $\cdot$ c - d $\cdot$ h $\cdot$ c - g $\cdot$ b $\cdot$ 1 - 1 $\cdot$ f $\cdot$ 1
= 1 + d $\cdot$ b $\cdot$ h + g $\cdot$ f $\cdot$ c - d $\cdot$ h $\cdot$ c - g $\cdot$ b - f

Case 2: When the off-diagonal elements are all 1
$\begin{vmatrix}
0 & 1 & 1 \
1 & 0 & 1 \
1 & 1 & 0
\end{vmatrix}$
= 0 $\cdot$ 0 $\cdot$ 0 + 1 $\cdot$ 1 $\cdot$ 1 + 1 $\cdot$ 1 $\cdot$ 1 - 1 $\cdot$ 1 $\cdot$ 1 - 1 $\cdot$ 1 $\cdot$ 1 - 0 $\cdot$ 1 $\cdot$ 1
= 2

Step 3: From the above cases, we can see that the maximum value of the determinant is 2. Therefore, the correct option is (a) 2.

Thus, the greatest value of the determinant is 2 when the off-diagonal elements are all 1.

Answer

2

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