Given `A = [a_(y)]_(2 xx 2) "where " a_(y) = {{:((|-3i + j|)/(2) if i ne j ),((i + j)^(2) If i = j ):} " then " a_(11) + a_(21) = `

Options

A. `6 . 5 `
B. `4 . 5 `
C. 20
D. 3

Answer

To solve the problem, we need to find the values of \( a_{11} \) and \( a_{21} \) based on the given definitions of the matrix elements.

### Step 1: Identify the values of \( a_{11} \) and \( a_{21} \).

The matrix \( A \) is defined as follows:
- \( a_{ij} = \frac{|-3i + j|}{2} \) if \( i \neq j \)
- \( a_{ij} = (i + j)^2 \) if \( i = j \)

### Step 2: Calculate \( a_{11} \).

Since \( a_{11} \) corresponds to \( i = 1 \) and \( j = 1 \), we use the second case of the definition:

\[ a_{11} = (1 + 1)^2 \]

Calculating this gives:

\[ a_{11} = 2^2 = 4 \]

### Step 3: Calculate \( a_{21} \).

For \( a_{21} \), we have \( i = 2 \) and \( j = 1 \), so we use the first case of the definition since \( i \neq j \):

\[ a_{21} = \frac{|-3(2) + 1|}{2} \]

Calculating the expression inside the absolute value:

\[ -3(2) + 1 = -6 + 1 = -5 \]

Now, taking the absolute value:

\[ |-5| = 5 \]

Now substituting back into the equation for \( a_{21} \):

\[ a_{21} = \frac{5}{2} \]

### Step 4: Calculate \( a_{11} + a_{21} \).

Now we can find the sum:

\[ a_{11} + a_{21} = 4 + \frac{5}{2} \]

To add these, we convert \( 4 \) into a fraction:

\[ 4 = \frac{8}{2} \]

Now we can add:

\[ a_{11} + a_{21} = \frac{8}{2} + \frac{5}{2} = \frac{8 + 5}{2} = \frac{13}{2} \]

### Final Result

Thus, the final result is:

\[ a_{11} + a_{21} = \frac{13}{2} = 6.5 \]