If `A=[{:(1,2,3), (0,4,2),(0,0,6):}]`, then the minor of the element `a_(31)` is

Options

A. 0
B. `-8`
C. 4
D. `-12`

Answer

To find the minor of the element \( a_{31} \) in the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 2 \\ 0 & 0 & 6 \end{pmatrix} \), we will follow these steps:

### Step 1: Identify the Element
The element \( a_{31} \) refers to the element in the 3rd row and 1st column of the matrix \( A \). In this case, \( a_{31} = 0 \).

### Step 2: Remove the Row and Column
To find the minor of \( a_{31} \), we need to remove the 3rd row and the 1st column from the matrix \( A \). The remaining elements will form a new 2x2 matrix.

Removing the 3rd row and 1st column, we have:
\[ \begin{pmatrix} 2 & 3 \\ 4 & 2 \end{pmatrix} \]

### Step 3: Calculate the Determinant
Next, we calculate the determinant of the resulting 2x2 matrix:
\[ \text{Determinant} = (2)(2) - (3)(4) \]
Calculating this gives:
\[ = 4 - 12 = -8 \]

### Step 4: Conclusion
Thus, the minor of the element \( a_{31} \) is \( -8 \).

### Summary of Solution
The minor of the element \( a_{31} \) in the matrix \( A \) is \( -8 \).

---