Sure, let's find the cofactor of the element [tex]\(a_{32}\)[/tex] in the given matrix [tex]\(\Delta\)[/tex].
Given the matrix
[tex]\[
\Delta = \begin{vmatrix}
1 & 3 & -2 \\
4 & -5 & 6 \\
3 & 5 & 2
\end{vmatrix},
\][/tex]
we need to determine the cofactor of the entry at the 3rd row and 2nd column, which is [tex]\(a_{32} = 5\)[/tex].
### Step 1: Find Minor of [tex]\(a_{32}\)[/tex]
To find the minor of [tex]\(a_{32}\)[/tex], we need to remove the 3rd row and 2nd column from the original matrix. The remaining 2x2 matrix is:
[tex]\[
\begin{vmatrix}
1 & -2 \\
4 & 6
\end{vmatrix}.
\][/tex]
### Step 2: Calculate Determinant of the Minor
Next, find the determinant of this 2x2 matrix:
[tex]\[
\text{Det} = (1 \cdot 6) - (-2 \cdot 4) = 6 - (-8) = 6 + 8 = 14.
\][/tex]
So, the minor of [tex]\(a_{32}\)[/tex] is [tex]\(14\)[/tex].
### Step 3: Calculate Cofactor of [tex]\(a_{32}\)[/tex]
The cofactor is given by multiplying the minor by [tex]\((-1)^{i+j}\)[/tex], where [tex]\(i\)[/tex] and [tex]\(j\)[/tex] are the row and column indices of the element. Here, [tex]\(i = 3\)[/tex] and [tex]\(j = 2\)[/tex]:
[tex]\[
\text{Cofactor of } a_{32} = (-1)^{3+2} \times \text{Minor of } a_{32}.
\][/tex]
Calculate the exponent:
[tex]\[
(-1)^{3+2} = (-1)^5 = -1.
\][/tex]
Thus, the cofactor of [tex]\(a_{32}\)[/tex] is:
[tex]\[
\text{Cofactor of } a_{32} = -1 \times 14 = -14.
\][/tex]
### Conclusion
The cofactor of [tex]\(a_{32}\)[/tex] in the matrix [tex]\(\Delta\)[/tex] is [tex]\(-14\)[/tex].
