Find the minor of element 6 in the determinant $\Delta=\left|\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right|$

Step 1: Identify the element and its position

The given determinant is a $3 \times 3$ matrix. We need to find the minor of the element 6. By inspecting the matrix, we can see that the element 6 is located in the second row and third column.

Step 2: Define the minor

The minor of an element $a_{ij}$, denoted as $M_{ij}$, is the determinant of the submatrix obtained by deleting the $i$ -th row and $j$ -th column containing the element.

Step 3: Form the submatrix

Since element 6 is $a_{23}$ (second row, third column), we delete the second row and the third column from the original determinant. The remaining elements form a $2 \times 2$ submatrix.

Step 4: Calculate the determinant of the submatrix

To find the minor $M_{23}$, we calculate the determinant of the $2 \times 2$ submatrix. The determinant of a $2 \times 2$ matrix $\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$ is given by $ad - bc$.

Step 5: Simplify the result

Performing the multiplication and subtraction, we find the value of the minor $M_{23}$.

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