What should be the value of x so that the matrix `({:(2,4),(-8,x):})` does not have an inverse ?

Options

A. 16
B. -16
C. 8
D. -8

Explanation

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

\[ A = \begin{pmatrix} 2 & 4 \\ -8 & x \end{pmatrix} \]

does not have an inverse, we need to find when the determinant of the matrix \( A \) is equal to zero. The determinant of a \( 2 \times 2 \) matrix

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

is given by the formula:

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

For our matrix \( A \), we have:

- \( a = 2 \)
- \( b = 4 \)
- \( c = -8 \)
- \( d = x \)

Now, we can calculate the determinant:

\[ \text{det}(A) = (2)(x) - (4)(-8) \]

Calculating the second term:

\[ \text{det}(A) = 2x + 32 \]

To find the value of \( x \) for which the matrix does not have an inverse, we set the determinant equal to zero:

\[ 2x + 32 = 0 \]

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

1. Subtract 32 from both sides:

\[ 2x = -32 \]

2. Divide both sides by 2:

\[ x = -16 \]

Thus, the value of \( x \) should be \( -16 \) for the matrix to not have an inverse.

### Summary of Steps:
1. Identify the matrix and its elements.
2. Use the determinant formula for a \( 2 \times 2 \) matrix.
3. Set the determinant equal to zero.
4. Solve for \( x \).