If `A=[{:(1,2),(4,2):}]`, then the value of `k=`_________if `|2A|=k|A|`.

Answer

To solve the problem, we need to find the value of \( k \) such that \( |2A| = k|A| \) for the matrix \( A = \begin{pmatrix} 1 & 2 \\ 4 & 2 \end{pmatrix} \).

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

\[ |A| = ad - bc \]

For our matrix \( A \):

\[ |A| = (1)(2) - (2)(4) = 2 - 8 = -6 \]

### Step 2: Calculate the matrix \( 2A \)

To find \( 2A \), we multiply each element of matrix \( A \) by 2:

\[ 2A = 2 \cdot \begin{pmatrix} 1 & 2 \\ 4 & 2 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 8 & 4 \end{pmatrix} \]

### Step 3: Calculate the determinant of matrix \( 2A \)

Now, we calculate the determinant of \( 2A \):

\[ |2A| = (2)(4) - (4)(8) = 8 - 32 = -24 \]

### Step 4: Set up the equation \( |2A| = k|A| \)

According to the problem, we have:

\[ |2A| = k |A| \]

Substituting the values we calculated:

\[ -24 = k(-6) \]

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

To find \( k \), we rearrange the equation:

\[ k = \frac{-24}{-6} = 4 \]

### Conclusion

Thus, the value of \( k \) is:

\[ \boxed{4} \]