For what value of x,`A=[(2(x+1),2x),(x,x-2)]` is a singular matrix.

Options

A. `-2`
B. `-3`
C. `-1`
D. 0

Answer

To find the value of \( x \) for which the matrix

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

is a singular matrix, we need to compute the determinant of the matrix and set it to zero. A matrix is singular if its determinant is equal to zero.

### Step 1: Calculate the determinant of matrix A.

The determinant of a 2x2 matrix

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

is given by the formula:

\[ \text{det}(A) = ad - bc \]

For our matrix \( A \):

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

Thus, the determinant can be calculated as:

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

### Step 2: Expand the determinant expression.

Now, we will expand the determinant:

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

This simplifies to:

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

### Step 3: Further simplify the expression.

Now, distribute the 2:

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

Combining like terms results in:

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

### Step 4: Set the determinant to zero.

To find the value of \( x \) for which the matrix is singular, we set the determinant equal to zero:

\[ -2x - 4 = 0 \]

### Step 5: Solve for x.

Now, we can solve for \( x \):

\[ -2x = 4 \]

Dividing both sides by -2 gives:

\[ x = -2 \]

### Conclusion

Thus, the value of \( x \) for which the matrix \( A \) is singular is

\[ \boxed{-2} \]