Step 1: Understanding the Question:
The problem asks us to calculate the determinant of a given \( 2 \times 2 \) square matrix \( A \).
The determinant of a matrix is a scalar value that provides important algebraic information about the matrix.
Step 2: Key Formula or Approach:
For a general \( 2 \times 2 \) matrix:
\[ A = \begin{bmatrix}a & b
c & d\end{bmatrix} \]
The determinant, denoted as \( |A| \) or \( \det(A) \), is calculated using the formula:
\[ |A| = ad - bc \]
We multiply the diagonal elements and subtract the product of the off-diagonal elements.
Step 3: Detailed Explanation:
Let us identify the components of the given matrix \( A = \begin{bmatrix}2 & 1
3 & 4\end{bmatrix} \):
- \( a = 2 \) (element at first row, first column)
- \( b = 1 \) (element at first row, second column)
- \( c = 3 \) (element at second row, first column)
- \( d = 4 \) (element at second row, second column)
Now, apply these values to the determinant formula:
\[ |A| = (2 \cdot 4) - (1 \cdot 3) \]
First, calculate the product of the principal diagonal elements:
\[ 2 \cdot 4 = 8 \]
Next, calculate the product of the secondary diagonal elements:
\[ 1 \cdot 3 = 3 \]
Now, subtract the secondary diagonal product from the principal diagonal product:
\[ |A| = 8 - 3 = 5 \]
Let us analyze why other options are incorrect:
- Option (B) \( 8 \) is incorrect because it is only the product of the principal diagonal, forgetting to subtract the other product.
- Option (C) \( 11 \) is incorrect because it adds the two products instead of subtracting them (\( 8 + 3 = 11 \)).
- Option (D) \( 13 \) is incorrect because of an arithmetic calculation error.
Step 4: Final Answer:
The determinant of the matrix \( A \) is \( 5 \), which corresponds to Option (A).