The value of x so that the matrix `[[2(x+1), 2x],[x, x-2]]` is singular, is :

Options

A. `-2`
B. `-1`
C. 1
D. 2

Answer

To find the value of \( x \) such that the matrix

\[ \begin{pmatrix} 2(x+1) & 2x \\ x & x-2 \end{pmatrix} \]

is singular, we need to calculate the determinant of the matrix and set it equal to zero.

### Step 1: Write the determinant of the matrix

The determinant of a \( 2 \times 2 \) matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is given by the formula:

\[ \text{det} = ad - bc \]

For our matrix, we have:

- \( a = 2(x+1) \)
- \( b = 2x \)
- \( c = x \)
- \( d = x-2 \)

Thus, the determinant can be calculated as:

\[ \text{det} = 2(x+1)(x-2) - (2x)(x) \]

### Step 2: Expand the determinant expression

Now, we will expand the expression:

\[ \text{det} = 2(x^2 - 2x + x - 2) - 2x^2 \]

This simplifies to:

\[ \text{det} = 2(x^2 - x - 2) - 2x^2 \]

### Step 3: Combine like terms

Now, we combine the terms:

\[ \text{det} = 2x^2 - 2x - 4 - 2x^2 \]

This simplifies to:

\[ \text{det} = -2x - 4 \]

### Step 4: Set the determinant to zero

For the matrix to be singular, we set the determinant equal to zero:

\[ -2x - 4 = 0 \]

### Step 5: Solve for \( x \)

Now, we solve for \( x \):

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

### Conclusion

Thus, the value of \( x \) such that the matrix is singular is

\[ \boxed{-2} \]