To find the determinant of the matrix
[tex]\[
\begin{vmatrix}
1+x & 1 & 1 \\
1 & 1+y & 1 \\
1 & 1 & 1+2
\end{vmatrix},
\][/tex]
we can use the cofactor expansion method. We'll expand along the first row. The formula for the determinant of a [tex]\(3 \times 3\)[/tex] matrix
[tex]\[
\begin{vmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{vmatrix}
\][/tex]
is given by:
[tex]\[
\text{det} = a \cdot \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \cdot \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \cdot \begin{vmatrix} d & e \\ g & h \end{vmatrix}.
\][/tex]
Here,
[tex]\( a = 1 + x \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = 1 \)[/tex], [tex]\( d = 1 \)[/tex], [tex]\( e = 1 + y \)[/tex], [tex]\( f = 1 \)[/tex], [tex]\( g = 1 \)[/tex], [tex]\( h = 1 \)[/tex], [tex]\( i = 3 \)[/tex] (since [tex]\( 1 + 2 = 3 \)[/tex]).
Applying this to our matrix:
[tex]\[
\text{det} = (1 + x) \cdot \begin{vmatrix} 1 + y & 1 \\ 1 & 3 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ 1 & 3 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 1 + y \\ 1 & 1 \end{vmatrix}
\][/tex]
Now we need to compute the [tex]\(2 \times 2\)[/tex] determinants:
1. [tex]\(\begin{vmatrix} 1 + y & 1 \\ 1 & 3 \end{vmatrix}\)[/tex]:
[tex]\[
= (1 + y) \cdot 3 - 1 \cdot 1 = 3 + 3y - 1 = 2 + 3y.
\][/tex]
2. [tex]\(\begin{vmatrix} 1 & 1 \\ 1 & 3 \end{vmatrix}\)[/tex]:
[tex]\[
= 1 \cdot 3 - 1 \cdot 1 = 3 - 1 = 2.
\][/tex]
3. [tex]\(\begin{vmatrix} 1 & 1 + y \\ 1 & 1 \end{vmatrix}\)[/tex]:
[tex]\[
= 1 \cdot 1 - (1 + y) \cdot 1 = 1 - 1 - y = -y.
\][/tex]
Putting these values into our determinant formula:
[tex]\[
\text{det} = (1 + x) \cdot (2 + 3y) - 1 \cdot 2 + 1 \cdot (-y)
\][/tex]
Expanding each term:
[tex]\[
= (1 + x)(2 + 3y) - 2 - y
\][/tex]
[tex]\[
= (1 \cdot 2 + 1 \cdot 3y + x \cdot 2 + x \cdot 3y) - 2 - y
\][/tex]
[tex]\[
= (2 + 3y + 2x + 3xy) - 2 - y
\][/tex]
Simplifying further, we combine like terms:
[tex]\[
= 2 + 3y + 2x + 3xy - 2 - y
\][/tex]
[tex]\[
= (2 - 2) + (3y - y) + 2x + 3xy
\][/tex]
[tex]\[
= 2x + 2y + 3xy
\][/tex]
Thus, the determinant is:
[tex]\[
\boxed{2x + 2y + 3xy}.
\][/tex]
