To find the cofactors of each entry in the first row of the matrix [tex]\( A = \begin{pmatrix} 3 & 1 & 4 \\ 1 & -4 & 7 \\ 6 & 3 & -2 \end{pmatrix} \)[/tex], we need to first understand the concept of the cofactor.
For a given element [tex]\( A[i][j] \)[/tex] in matrix [tex]\( A \)[/tex], the cofactor is determined by:
1. Deleting the [tex]\(i\)[/tex]-th row and [tex]\(j\)[/tex]-th column to form the minor matrix.
2. Taking the determinant of the resulting minor matrix.
3. Multiplying the determinant by [tex]\( (-1)^{i+j} \)[/tex] to apply the checkerboard pattern of signs.
Let's do this for each entry in the first row of the matrix [tex]\( A \)[/tex].
### Entry [tex]\( A[0][0] \)[/tex] - which is 3
1. Remove the 0-th row and 0-th column:
[tex]\[
\text{Minor matrix} = \begin{pmatrix} -4 & 7 \\ 3 & -2 \end{pmatrix}
\][/tex]
2. Find the determinant of the minor matrix:
[tex]\[
\text{Determinant} = (-4)(-2) - (7)(3) = 8 - 21 = -13
\][/tex]
3. Apply the sign pattern [tex]\( (-1)^{0+0} = 1 \)[/tex]:
[tex]\[
\text{Cofactor} = 1 \times (-13) = -13
\][/tex]
So, [tex]\( c_{11} = -13 \)[/tex].
### Entry [tex]\( A[0][1] \)[/tex] - which is 1
1. Remove the 0-th row and 1-st column:
[tex]\[
\text{Minor matrix} = \begin{pmatrix} 1 & 7 \\ 6 & -2 \end{pmatrix}
\][/tex]
2. Find the determinant of the minor matrix:
[tex]\[
\text{Determinant} = (1)(-2) - (7)(6) = -2 - 42 = -44
\][/tex]
3. Apply the sign pattern [tex]\( (-1)^{0+1} = -1 \)[/tex]:
[tex]\[
\text{Cofactor} = -1 \times (-44) = 44
\][/tex]
So, [tex]\( c_{12} = 44 \)[/tex].
### Entry [tex]\( A[0][2] \)[/tex] - which is 4
1. Remove the 0-th row and 2-nd column:
[tex]\[
\text{Minor matrix} = \begin{pmatrix} 1 & -4 \\ 6 & 3 \end{pmatrix}
\][/tex]
2. Find the determinant of the minor matrix:
[tex]\[
\text{Determinant} = (1)(3) - (-4)(6) = 3 + 24 = 27
\][/tex]
3. Apply the sign pattern [tex]\( (-1)^{0+2} = 1 \)[/tex]:
[tex]\[
\text{Cofactor} = 1 \times 27 = 27
\][/tex]
So, [tex]\( c_{13} = 27 \)[/tex].
So, the cofactors for the entries in the first row of the matrix [tex]\( A \)[/tex] are:
[tex]\[
\begin{array}{l}
c_{11} = -13 \\
c_{12} = 44 \\
c_{13} = 27
\end{array}
\][/tex]
