Find the value of x for which the matrix `A=[(2//x,-1,2),(1,x,2x^(2)),(1,1//x,2)]` is singular.

Explanation

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

\[ A = \begin{pmatrix} \frac{2}{x} & -1 & 2 \\ 1 & x & 2x^2 \\ 1 & \frac{1}{x} & 2 \end{pmatrix} \]

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

### Step 1: Calculate the Determinant of Matrix A

The determinant of a 3x3 matrix

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

is given by the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

- \( a = \frac{2}{x} \)
- \( b = -1 \)
- \( c = 2 \)
- \( d = 1 \)
- \( e = x \)
- \( f = 2x^2 \)
- \( g = 1 \)
- \( h = \frac{1}{x} \)
- \( i = 2 \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = \frac{2}{x} \left( x \cdot 2 - 2x^2 \cdot \frac{1}{x} \right) - (-1) \left( 1 \cdot 2 - 2x^2 \cdot 1 \right) + 2 \left( 1 \cdot \frac{1}{x} - x \cdot 1 \right) \]

### Step 2: Simplify the Determinant Expression

Calculating each term:

1. First term:
\[ \frac{2}{x} (2x - 2x) = \frac{2}{x} \cdot 0 = 0 \]

2. Second term:
\[ +1 (2 - 2x^2) = 2 - 2x^2 \]

3. Third term:
\[ 2 \left( \frac{1}{x} - x \right) = 2 \left( \frac{1 - x^2}{x} \right) = \frac{2(1 - x^2)}{x} \]

Combining these, we have:

\[ \text{det}(A) = 0 + (2 - 2x^2) + \frac{2(1 - x^2)}{x} \]

### Step 3: Set the Determinant Equal to Zero

Now we set the determinant equal to zero:

\[ 2 - 2x^2 + \frac{2(1 - x^2)}{x} = 0 \]

### Step 4: Multiply through by \( x \) to eliminate the fraction

Multiplying the entire equation by \( x \) (assuming \( x \neq 0 \)) gives:

\[ 2x - 2x^3 + 2(1 - x^2) = 0 \]

This simplifies to:

\[ 2x - 2x^3 + 2 - 2x^2 = 0 \]

Rearranging gives:

\[ -2x^3 + 2x - 2x^2 + 2 = 0 \]

Dividing through by -2:

\[ x^3 - x^2 - x - 1 = 0 \]

### Step 5: Factor the Polynomial

We can factor this polynomial. We can try \( x + 1 \) as a factor:

Using synthetic division or polynomial long division, we find:

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

### Step 6: Solve for x

Setting each factor to zero gives:

1. \( x + 1 = 0 \) → \( x = -1 \)
2. \( x^2 - 2 = 0 \) → \( x = \pm \sqrt{2} \)

### Final Answer

The values of \( x \) for which the matrix \( A \) is singular are:

\[ x = -1, \quad x = \sqrt{2}, \quad x = -\sqrt{2} \]