What is the value of the minor of the element 9 in the determinant `|{:(10,19,2),(0,13,1),(9,24,2):}|` ?

Options

A. -9
B. -7
C. 7
D. 0

Explanation

To find the value of the minor of the element 9 in the determinant

\[ \begin{vmatrix} 10 & 19 & 2 \\ 0 & 13 & 1 \\ 9 & 24 & 2 \end{vmatrix} \]

we follow these steps:

### Step 1: Identify the position of the element
The element 9 is located in the third row and the first column of the matrix.

### Step 2: Remove the row and column
To find the minor of the element 9, we need to remove the row and column that contain this element. This means we will remove the third row and the first column.

The remaining elements form a 2x2 matrix:

\[ \begin{vmatrix} 19 & 2 \\ 13 & 1 \end{vmatrix} \]

### Step 3: Calculate the determinant of the 2x2 matrix
The determinant of a 2x2 matrix

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

is calculated using the formula \( ad - bc \).

For our matrix:

- \( a = 19 \)
- \( b = 2 \)
- \( c = 13 \)
- \( d = 1 \)

So, we calculate:

\[ \text{Determinant} = (19 \times 1) - (2 \times 13) \]

### Step 4: Perform the multiplication
Calculating the products:

\[ 19 \times 1 = 19 \]
\[ 2 \times 13 = 26 \]

### Step 5: Subtract the results
Now we subtract the second product from the first:

\[ 19 - 26 = -7 \]

### Final Result
Thus, the value of the minor of the element 9 is \(-7\).

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