`"If"|[5, 3, -1], [-7, x, 2], [9, 6, -2]|=0 " then " value of -x = ?`

Options

A. 0
B. 6
C. `-6`
D. 9

Explanation

To solve the determinant equation given by

\[ \begin{vmatrix} 5 & 3 & -1 \\ -7 & x & 2 \\ 9 & 6 & -2 \end{vmatrix} = 0, \]

we will calculate the determinant step by step and find the value of \( -x \).

### Step 1: Calculate the determinant

The determinant of a 3x3 matrix

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \]

is calculated using the formula:

\[ a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}. \]

For our determinant:

\[ 5 \begin{vmatrix} x & 2 \\ 6 & -2 \end{vmatrix} - 3 \begin{vmatrix} -7 & 2 \\ 9 & -2 \end{vmatrix} - 1 \begin{vmatrix} -7 & x \\ 9 & 6 \end{vmatrix}. \]

### Step 2: Calculate the smaller 2x2 determinants

1. For \( 5 \begin{vmatrix} x & 2 \\ 6 & -2 \end{vmatrix} \):

\[ \begin{vmatrix} x & 2 \\ 6 & -2 \end{vmatrix} = x(-2) - 2(6) = -2x - 12. \]

Thus,

\[ 5(-2x - 12) = -10x - 60. \]

2. For \( -3 \begin{vmatrix} -7 & 2 \\ 9 & -2 \end{vmatrix} \):

\[ \begin{vmatrix} -7 & 2 \\ 9 & -2 \end{vmatrix} = (-7)(-2) - (2)(9) = 14 - 18 = -4. \]

Thus,

\[ -3(-4) = 12. \]

3. For \( -1 \begin{vmatrix} -7 & x \\ 9 & 6 \end{vmatrix} \):

\[ \begin{vmatrix} -7 & x \\ 9 & 6 \end{vmatrix} = (-7)(6) - (x)(9) = -42 - 9x. \]

Thus,

\[ -1(-42 - 9x) = 42 + 9x. \]

### Step 3: Combine all parts

Now, we combine all the parts together:

\[ -10x - 60 + 12 + 42 + 9x = 0. \]

This simplifies to:

\[ -x - 6 = 0. \]

### Step 4: Solve for \( x \)

Rearranging gives:

\[ -x = 6. \]

Thus,

\[ x = -6. \]

### Step 5: Find \( -x \)

Finally, we need to find \( -x \):

\[ -x = 6. \]

### Final Answer

The value of \( -x \) is \( 6 \).

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