If `Delta=|{:(5,3,8),(2,0,1),(1,2,3):}|` , write :
`(i)` the minor of the element `a_(23)`
`(ii)` the co-factor of the element `a_(32)`.

Answer

To solve the given problem, we will find the minor of the element \( a_{23} \) and the cofactor of the element \( a_{32} \) from the determinant \( \Delta = \begin{vmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{vmatrix} \).

### Step 1: Identify the elements
The element \( a_{23} \) is located in the 2nd row and 3rd column of the determinant, which is \( 1 \). The element \( a_{32} \) is located in the 3rd row and 2nd column, which is \( 2 \).

### Step 2: Calculate the Minor of \( a_{23} \)
To find the minor of \( a_{23} \), we need to eliminate the 2nd row and 3rd column from the determinant.

The remaining elements are:
\[ \begin{vmatrix} 5 & 3 \\ 1 & 2 \end{vmatrix} \]

Now, we calculate this 2x2 determinant:
\[ \text{Minor of } a_{23} = (5 \cdot 2) - (3 \cdot 1) = 10 - 3 = 7 \]

### Step 3: Calculate the Cofactor of \( a_{32} \)
To find the cofactor of \( a_{32} \), we first calculate the minor of \( a_{32} \) and then apply the sign factor.

The minor of \( a_{32} \) is found by eliminating the 3rd row and 2nd column:
\[ \begin{vmatrix} 5 & 8 \\ 2 & 3 \end{vmatrix} \]

Calculating this 2x2 determinant:
\[ \text{Minor of } a_{32} = (5 \cdot 3) - (8 \cdot 2) = 15 - 16 = -1 \]

Now, we apply the sign factor for the cofactor. The sign factor is given by \( (-1)^{m+n} \), where \( m \) is the row index and \( n \) is the column index. For \( a_{32} \), \( m = 3 \) and \( n = 2 \):
\[ (-1)^{3+2} = (-1)^{5} = -1 \]

Thus, the cofactor of \( a_{32} \) is:
\[ \text{Cofactor of } a_{32} = (-1) \cdot (-1) = 1 \]

### Final Answers
(i) The minor of the element \( a_{23} \) is \( 7 \).

(ii) The cofactor of the element \( a_{32} \) is \( 1 \).