Find |A| If `A = |(x^2, 2x), (x^4 , 5x)|`

Answer

To find the determinant |A| where

\[ A = \begin{pmatrix} x^2 & 2x \\ x^4 & 5x \end{pmatrix} \]

we will follow the steps for calculating the determinant of a 2x2 matrix.

### Step-by-step Solution:

1. **Write the determinant formula for a 2x2 matrix:**
The determinant of a matrix
\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]
is given by the formula:
\[ |A| = ad - bc \]

2. **Identify the elements of the matrix A:**
For our matrix
\[ A = \begin{pmatrix} x^2 & 2x \\ x^4 & 5x \end{pmatrix} \]
we can identify:
- \( a = x^2 \)
- \( b = 2x \)
- \( c = x^4 \)
- \( d = 5x \)

3. **Substitute the values into the determinant formula:**
Now, we substitute the identified values into the determinant formula:
\[ |A| = (x^2)(5x) - (2x)(x^4) \]

4. **Calculate the products:**
- First term: \( (x^2)(5x) = 5x^3 \)
- Second term: \( (2x)(x^4) = 2x^5 \)

5. **Combine the terms:**
Now we combine the results:
\[ |A| = 5x^3 - 2x^5 \]

6. **Factor out the common term:**
We can factor out \( x^3 \) from both terms:
\[ |A| = x^3(5 - 2x^2) \]

### Final Result:
Thus, the determinant |A| is:
\[ |A| = x^3(5 - 2x^2) \]