If `Delta = |(5,3,8),(2,0,1),(1,2,3)|`, the write the minor of the element `a_(23)`. a) 7 b) -7 c) 4 d)8

Options

A. 7
B. `-7`
C. 4
D. 8

Answer

To find the minor of the element \( a_{23} \) in the determinant \( \Delta = |(5,3,8),(2,0,1),(1,2,3)| \), we follow these steps:

### Step 1: Identify the element for which we need to find the minor
The element \( a_{23} \) refers to the element in the 2nd row and 3rd column of the matrix. In our case, this element is \( 1 \) (from the second row: \( (2, 0, 1) \)).

### Step 2: Remove the row and column of the element
To find the minor of \( a_{23} \), we need to remove the 2nd row and the 3rd column from the matrix. The remaining elements will form a new 2x2 matrix.

The original matrix is:
\[ \begin{pmatrix} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{pmatrix} \]

After removing the 2nd row and 3rd column, we are left with:
\[ \begin{pmatrix} 5 & 3 \\ 1 & 2 \end{pmatrix} \]

### Step 3: Calculate the determinant of the new 2x2 matrix
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as \( ad - bc \).

For our matrix:
\[ \begin{pmatrix} 5 & 3 \\ 1 & 2 \end{pmatrix} \]
The determinant is calculated as follows:
\[ \text{Determinant} = (5 \cdot 2) - (3 \cdot 1) = 10 - 3 = 7 \]

### Step 4: Conclusion
The minor of the element \( a_{23} \) is \( 7 \).

Thus, the answer is \( 7 \).

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