If `A = [{:(3, 4),(1,2):}],` then `|2A|=`

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the determinant of the matrix \(2A\), where \(A\) is given as:

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

### Step 1: Multiply the matrix \(A\) by 2

First, we multiply each element of matrix \(A\) by 2:

\[ 2A = 2 \times \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 2 \times 3 & 2 \times 4 \\ 2 \times 1 & 2 \times 2 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 2 & 4 \end{pmatrix} \]

### Step 2: Write down the matrix \(2A\)

Now we have:

\[ 2A = \begin{pmatrix} 6 & 8 \\ 2 & 4 \end{pmatrix} \]

### Step 3: Find the determinant of the matrix \(2A\)

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

\[ \text{det}(A) = ad - bc \]

For our matrix \(2A\):

- \(a = 6\)
- \(b = 8\)
- \(c = 2\)
- \(d = 4\)

Now, substitute these values into the determinant formula:

\[ \text{det}(2A) = (6 \times 4) - (8 \times 2) \]

Calculating each term:

\[ 6 \times 4 = 24 \]
\[ 8 \times 2 = 16 \]

Now, subtract the second term from the first:

\[ \text{det}(2A) = 24 - 16 = 8 \]

### Final Answer

Thus, the determinant \(|2A| = 8\).

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