Question

If \[ A = \begin{bmatrix}2 & 1 \\ 3 & 4\end{bmatrix} \] then find \( |A| \).

• A. \( 5 \)
• B. \( 8 \)
• C. \( 11 \)
• D. \( 13 \)

Hint:

Always remember the sign convention when calculating determinants.
The principal diagonal product has a positive sign, while the secondary diagonal product is subtracted.
Double-check simple multiplication and subtraction steps, as these are common areas for careless errors.

Answer 1

The Correct Option is A

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 by \(|A|\) or \(\det(A)\), is calculated using:

\[ |A| = ad - bc \]

We multiply the diagonal elements and subtract the product of the off-diagonal elements.

Step 3: Detailed Explanation:

Consider the matrix:

\[ A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix} \]

Identify the matrix elements:

\(a = 2\)  (first row, first column) 
\(b = 1\)   (first row, second column) 
\(c = 3\)   (second row, first column) 
\(d = 4\)   (second row, second column)

Substitute these values into the determinant formula:

\[ |A| = (2 \cdot 4) - (1 \cdot 3) \]

First, calculate the product of the principal diagonal:

\[ 2 \cdot 4 = 8 \]

Next, calculate the product of the secondary diagonal:

\[ 1 \cdot 3 = 3 \]

Now subtract:

\[ |A| = 8 - 3 = 5 \] 
 

Why the Other Options are Incorrect:

Option (B): \(8\) 
Only the product of the principal diagonal is considered, while the second product is not subtracted.

Option (C): \(11\) 
The two products are added instead of subtracted: \(8 + 3 = 11\).

Option (D): \(13\) 
This results from an arithmetic mistake.

Step 4: Final Answer:

The determinant of matrix \(A\) is:

\[ \boxed{5} \]

Therefore, the correct answer is Option (A).