Find the value of `a_(23)+a_(32)` in the matrix
`A=[a_(ij)]_(3xx3)` where `a_(ij)={{:(|2i-j|,"if "igtj),(-i+2j+3,"if "iltj):}`

Answer

To find the value of \( a_{23} + a_{32} \) in the matrix \( A = [a_{ij}]_{3 \times 3} \), we will use the given conditions for \( a_{ij} \):

\[ a_{ij} = \begin{cases} |2i - j| & \text{if } i > j \\ - i + 2j + 3 & \text{if } i < j \end{cases} \]

### Step 1: Calculate \( a_{23} \)
Since \( i = 2 \) and \( j = 3 \), we have \( i < j \). Therefore, we will use the second condition:

\[ a_{23} = -2 + 2 \cdot 3 + 3 \]

Calculating this:

\[ a_{23} = -2 + 6 + 3 = 7 \]

### Step 2: Calculate \( a_{32} \)
Now, for \( a_{32} \), we have \( i = 3 \) and \( j = 2 \). Here, \( i > j \), so we will use the first condition:

\[ a_{32} = |2 \cdot 3 - 2| \]

Calculating this:

\[ a_{32} = |6 - 2| = |4| = 4 \]

### Step 3: Find \( a_{23} + a_{32} \)
Now we can find the sum:

\[ a_{23} + a_{32} = 7 + 4 = 11 \]

Thus, the final answer is:

\[ \boxed{11} \]