If `A=[a_(ij)]_(2xx2)" where "a_(ij)=2i+j," then :"A=`

Options

A. `{:[(3,4),(5,6)]:}`
B. `{:[(3,5),(4,6)]:}`
C. `{:[(3,5),(5,6)]:}`
D. `{:[(3,6),(5,4)]:}`

Answer

To find the matrix \( A = [a_{ij}]_{2 \times 2} \) where \( a_{ij} = 2i + j \), we will compute each element of the matrix step by step.

### Step 1: Define the matrix structure
The matrix \( A \) is a \( 2 \times 2 \) matrix, which means it has 2 rows and 2 columns. We can represent it as:
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]

### Step 2: Calculate \( a_{11} \)
To find \( a_{11} \), we set \( i = 1 \) and \( j = 1 \):
\[ a_{11} = 2(1) + 1 = 2 + 1 = 3 \]

### Step 3: Calculate \( a_{12} \)
Next, we calculate \( a_{12} \) by setting \( i = 1 \) and \( j = 2 \):
\[ a_{12} = 2(1) + 2 = 2 + 2 = 4 \]

### Step 4: Calculate \( a_{21} \)
Now, we calculate \( a_{21} \) by setting \( i = 2 \) and \( j = 1 \):
\[ a_{21} = 2(2) + 1 = 4 + 1 = 5 \]

### Step 5: Calculate \( a_{22} \)
Finally, we calculate \( a_{22} \) by setting \( i = 2 \) and \( j = 2 \):
\[ a_{22} = 2(2) + 2 = 4 + 2 = 6 \]

### Step 6: Construct the matrix
Now that we have all the elements, we can write the matrix \( A \):
\[ A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix} \]

### Final Answer
Thus, the matrix \( A \) is:
\[ A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix} \]