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To solve the problem, we need to find the absolute value of the determinant of the adjugate matrix of \( A \), denoted as \( | \text{adj} A | \).

Given the matrix:

\[
A = \begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix}
\]

1. **Step 1: Find the determinant of \( A \)**

   The determinant of \( A \), denoted as \( \det(A) \), is calculated as:

   \[
   \det(A) = (2 \times 5) - (-1 \times 4) = 10 + 4 = 14
   \]

2. **Step 2: Find the determinant of the adjugate of \( A \)**

   The determinant of the adjugate of a 2x2 matrix \( A \) is related to the determinant of \( A \) by the formula:

   \[
   |\text{adj}(A)| = |\det(A)|^{n-1}
   \]

   For a 2x2 matrix (\( n = 2 \)):

   \[
   |\text{adj}(A)| = |\det(A)|^{2-1} = |\det(A)| = |14| = 14
   \]

Therefore, \( |\text{adj}(A)| = 14 \).

**Answer: B. 14**

Would you like more details or have any other questions?

Here are some related questions to expand your understanding:

1. What is the adjugate of a matrix, and how is it calculated?
2. How do you find the inverse of a 2x2 matrix using its adjugate?
3. What are the properties of determinants that simplify calculations?
4. How does the determinant change if we swap rows or columns in a matrix?
5. What is the relationship between the determinant and the eigenvalues of a matrix?

**Tip:** Remember that for a 2x2 matrix, the adjugate is simply the matrix obtained by swapping the diagonal elements and changing the signs of the off-diagonal elements!

If A = [[2, -1], [4, 5]], then |adj(A)| is?

Solve the problem of finding the adjugate determinant |adj(A)| for matrix A = [[2, -1], [4, 5]]. This problem uses the determinant formula for 2x2 matrices and explores the relationship between the determinant and the adjugate matrix. Suitable for undergraduate students.

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