Question

The value of \( x \) that satisfies the equation:

\[ \begin{vmatrix} x & 1 & 1 \\ 2 & 2 & 0 \\ 1 & 0 & -2 \end{vmatrix} = 6 \]

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(-2\)
• E. \(-1\)

Hint:

To find the value of \( x \) in a determinant equation, first expand the determinant of the given matrix. Use the formula for the determinant of a \(3 \times 3\) matrix:

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

After expanding, solve for \( x \) by setting the result equal to the given value.

Answer 1

Answer: (E)

We are given the determinant equation:

\[ \begin{vmatrix} x & 1 & 1 \\ 2 & 2 & 0 \\ 1 & 0 & -2 \end{vmatrix} = 6 \]

We need to evaluate the determinant of this matrix.

The determinant of a \(3 \times 3\) matrix is calculated using the formula:

\[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For the given matrix:

\[ \begin{vmatrix} x & 1 & 1 \\ 2 & 2 & 0 \\ 1 & 0 & -2 \end{vmatrix} \]

We have \( a = x \), \( b = 1 \), \( c = 1 \), \( d = 2 \), \( e = 2 \), \( f = 0 \), \( g = 1 \), \( h = 0 \), and \( i = -2 \). Applying the determinant formula:

\[ = x \left( 2(-2) - 0(0) \right) - 1 \left( 2(-2) - 0(1) \right) + 1 \left( 2(0) - 2(1) \right) \]

\[ = x(-4) - 1(-4) + 1(-2) \]

\[ = -4x + 4 - 2 \]

\[ = -4x + 2 \]

Setting this equal to 6:

\[ -4x + 2 = 6 \]

\[ -4x = 4 \]

\[ x = -1 \]

Thus, the value of \( x \) is \( -1 \).

Therefore, the correct answer is option (E), \( x = -1 \).