To determine the minor and the cofactor of the element in the second row, third column of the given matrix [tex]\(\Delta = \begin{vmatrix} 2 & -2 & 3 \\ 1 & 4 & 5 \\ 2 & 1 & -3 \end{vmatrix}\)[/tex], follow these steps:
### Step 1: Extract the Sub-matrix for the Minor
First, we need to find the minor of the element located in the second row and third column of the matrix. To do this, remove the second row and the third column from the matrix to form a new [tex]\(2 \times 2\)[/tex] sub-matrix. The resulting sub-matrix is:
[tex]\[
\begin{vmatrix}
2 & -2 \\
2 & 1
\end{vmatrix}
\][/tex]
### Step 2: Calculate the Determinant of the Sub-matrix (Minor)
Next, calculate the determinant of this [tex]\(2 \times 2\)[/tex] sub-matrix. The determinant of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\)[/tex] is given by [tex]\(ad - bc\)[/tex].
In our case, the sub-matrix is:
[tex]\[
\begin{vmatrix}
2 & -2 \\
2 & 1
\end{vmatrix}
\][/tex]
So, let's calculate its determinant:
[tex]\[
(2 \times 1) - (-2 \times 2) = 2 + 4 = 6
\][/tex]
Thus, the minor of the element at the second row and third column is [tex]\(6\)[/tex].
### Step 3: Determine the Cofactor
The cofactor is calculated by multiplying the minor by [tex]\((-1)^{i+j}\)[/tex], where [tex]\(i\)[/tex] is the row number and [tex]\(j\)[/tex] is the column number of the element we are focusing on. In this case, for the element in the second row ([tex]\(i = 2\)[/tex]) and third column ([tex]\(j = 3\)[/tex]), we have:
[tex]\[
(-1)^{2+3} = (-1)^5 = -1
\][/tex]
Therefore, the cofactor is:
[tex]\[
\text{cofactor} = (-1) \times 6 = -6
\][/tex]
### Conclusion
For the given matrix [tex]\(\Delta = \begin{vmatrix} 2 & -2 & 3 \\ 1 & 4 & 5 \\ 2 & 1 & -3 \end{vmatrix}\)[/tex], the results are as follows:
- The sub-matrix used to calculate the minor is:
[tex]\[
\begin{vmatrix}
2 & -2 \\
2 & 1
\end{vmatrix}
\][/tex]
- The minor of the element in the second row, third column is [tex]\(6\)[/tex].
- The cofactor of the element in the second row, third column is [tex]\(-6\)[/tex].
