If `|[x,3, 6],[ 3, 6,x],[6,x,3]|=|[2,x,7],[x,7, 2],[ 7, 2,x]|=|[4, 5,x],[5,x,4],[x,4, 5]|=0,` then `x` is equal to a.`0` b. `-9` c. `3` d. none of these

Options

A. 0
B. -9
C. 3
D. none of these

Explanation

To solve the problem, we need to find the value of \( x \) such that the determinants of the three given matrices are all equal to zero. We can denote the determinants as follows:

1. \( D_1 = \begin{vmatrix} x & 3 & 6 \\ 3 & 6 & x \\ 6 & x & 3 \end{vmatrix} \)
2. \( D_2 = \begin{vmatrix} 2 & x & 7 \\ x & 7 & 2 \\ 7 & 2 & x \end{vmatrix} \)
3. \( D_3 = \begin{vmatrix} 4 & 5 & x \\ 5 & x & 4 \\ x & 4 & 5 \end{vmatrix} \)

We need to find \( x \) such that \( D_1 = D_2 = D_3 = 0 \).

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