To find the determinant of the matrix
[tex]\[
\begin{vmatrix}
a + x & -b & c \\
a & b + y & c \\
a & b & c + 2
\end{vmatrix},
\][/tex]
we will use the method of cofactor expansion (also known as Laplace expansion). We will expand along the first row.
The matrix is
[tex]\[
\begin{vmatrix}
a + x & -b & c \\
a & b + y & c \\
a & b & c + 2
\end{vmatrix}.
\][/tex]
### Step 1: Cofactor Expansion Along the First Row
The determinant is given by:
[tex]\[
\det(A) = (a + x) \cdot \begin{vmatrix}
b + y & c \\
b & c + 2
\end{vmatrix}
- (-b) \cdot \begin{vmatrix}
a & c \\
a & c + 2
\end{vmatrix}
+ c \cdot \begin{vmatrix}
a & b + y \\
a & b
\end{vmatrix}.
\][/tex]
### Step 2: Compute the 2x2 Determinants
1. First Minor:
[tex]\[
\begin{vmatrix}
b + y & c \\
b & c + 2
\end{vmatrix}
= (b + y)(c + 2) - bc
= bc + 2b + yc + 2y - bc
= 2b + yc + 2y.
\][/tex]
2. Second Minor:
[tex]\[
\begin{vmatrix}
a & c \\
a & c + 2
\end{vmatrix}
= a(c + 2) - ac
= ac + 2a - ac
= 2a.
\][/tex]
3. Third Minor:
[tex]\[
\begin{vmatrix}
a & b + y \\
a & b
\end{vmatrix}
= a \cdot b - a \cdot (b + y)
= ab - ab - ay
= -ay.
\][/tex]
### Step 3: Substitute the Values Back into the Determinant Formula
Now substitute the computed minors back:
[tex]\[
\det(A) = (a + x)(2b + yc + 2y) - (-b)(2a) + c(-ay).
\][/tex]
### Step 4: Simplify the Expression
1. Expand First Term:
[tex]\[
(a + x)(2b + yc + 2y)
= a(2b + yc + 2y) + x(2b + yc + 2y)
= 2ab + ayc + 2ay + 2bx + xyc + 2xy.
\][/tex]
2. Simplify Second Term:
[tex]\[
- (-b)(2a)
= 2ab.
\][/tex]
3. Simplify Third Term:
[tex]\[
c(-ay)
= -acy.
\][/tex]
Putting these together:
[tex]\[
\det(A) = (2ab + ayc + 2ay + 2bx + xyc + 2xy) + 2ab - acy.
\][/tex]
Notice the [tex]\( + ayc \)[/tex] and [tex]\( - acy \)[/tex] cancel each other:
[tex]\[
\det(A) = 2ab + 2ab + 2ay + 2bx + xyc + 2xy
= 4ab + 2ay + 2bx + xyc + 2xy.
\][/tex]
Therefore, the determinant of the given matrix is:
[tex]\[
4ab + 2ay + 2bx + xyc + 2xy.
\][/tex]
