To find the cofactors of each entry in the first row of the given matrix [tex]\( A = \left[\begin{array}{ccc} 3 & 1 & 4 \\ 1 & -4 & 7 \\ 6 & 3 & -2 \end{array}\right] \)[/tex], we will denote the cofactors as [tex]\( AC_{ij} \)[/tex], where [tex]\( i \)[/tex] and [tex]\( j \)[/tex] represent the row and column indices, respectively.
Let's compute the cofactors for the first row.
### 1. [tex]\( AC_{11} \)[/tex]
To find [tex]\( AC_{11} \)[/tex], we need to eliminate the first row and the first column of matrix [tex]\( A \)[/tex], and then find the determinant of the resulting [tex]\( 2 \times 2 \)[/tex] matrix:
[tex]\[
\begin{vmatrix}
-4 & 7 \\
3 & -2
\end{vmatrix}
\][/tex]
The determinant of this [tex]\(2 \times 2\)[/tex] matrix is:
[tex]\[
(-4 \times -2) - (7 \times 3) = 8 - 21 = -13
\][/tex]
Therefore, the cofactor [tex]\( AC_{11} = -13 \)[/tex].
### 2. [tex]\( AC_{12} \)[/tex]
To find [tex]\( AC_{12} \)[/tex], we need to eliminate the first row and the second column of matrix [tex]\( A \)[/tex], and then find the determinant of the resulting [tex]\( 2 \times 2 \)[/tex] matrix:
[tex]\[
\begin{vmatrix}
1 & 7 \\
6 & -2
\end{vmatrix}
\][/tex]
The determinant of this [tex]\(2 \times 2\)[/tex] matrix is:
[tex]\[
(1 \times -2) - (7 \times 6) = -2 - 42 = -44
\][/tex]
Because [tex]\( AC_{12} \)[/tex] is in the second column of the first row, we need to take the negative of the determinant:
[tex]\[
AC_{12} = -(-44) = 44
\][/tex]
### 3. [tex]\( AC_{13} \)[/tex]
To find [tex]\( AC_{13} \)[/tex], we need to eliminate the first row and the third column of matrix [tex]\( A \)[/tex], and then find the determinant of the resulting [tex]\( 2 \times 2 \)[/tex] matrix:
[tex]\[
\begin{vmatrix}
1 & -4 \\
6 & 3
\end{vmatrix}
\][/tex]
The determinant of this [tex]\(2 \times 2\)[/tex] matrix is:
[tex]\[
(1 \times 3) - (-4 \times 6) = 3 + 24 = 27
\][/tex]
Therefore, the cofactor [tex]\( AC_{13} = 27 \)[/tex].
Putting it all together, we have:
[tex]\[
\begin{array}{l}
A C_{11} = -13 \\
A C_{12} = 44 \\
A C_{13} = 27
\end{array}
\][/tex]
