Find the value of 'x' if :
`|{:(-1,2),(4,8):}|=|{:(2,x),(x,-4):}|`

Answer

To solve for the value of \( x \) in the equation \( | \begin{pmatrix} -1 & 2 \\ 4 & 8 \end{pmatrix} | = | \begin{pmatrix} 2 & x \\ x & -4 \end{pmatrix} | \), we will calculate the determinants of both matrices and set them equal to each other.

### Step 1: Calculate the determinant of the left-hand side (LHS)

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

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

For the LHS matrix \( \begin{pmatrix} -1 & 2 \\ 4 & 8 \end{pmatrix} \):

\[ \text{det} = (-1)(8) - (2)(4) = -8 - 8 = -16 \]

### Step 2: Calculate the determinant of the right-hand side (RHS)

For the RHS matrix \( \begin{pmatrix} 2 & x \\ x & -4 \end{pmatrix} \):

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

### Step 3: Set the determinants equal to each other

Now we set the LHS equal to the RHS:

\[ -16 = -8 - x^2 \]

### Step 4: Solve for \( x^2 \)

To isolate \( x^2 \), we first add 8 to both sides:

\[ -16 + 8 = -x^2 \]
\[ -8 = -x^2 \]

Now, multiply both sides by -1:

\[ 8 = x^2 \]

### Step 5: Take the square root of both sides

Taking the square root gives us:

\[ x = \pm \sqrt{8} \]

We can simplify \( \sqrt{8} \):

\[ x = \pm 2\sqrt{2} \]

### Final Answer

Thus, the values of \( x \) are:

\[ x = 2\sqrt{2} \quad \text{or} \quad x = -2\sqrt{2} \]

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