To calculate the determinant of the matrix
[tex]\[
\left|\begin{array}{ccc}
1 & -1 & 3 \\
2 & 5 & 0 \\
-3 & 1 & 2
\end{array}\right|
\][/tex]
we'll use the cofactor expansion method (also known as Laplace expansion). Let's apply this method to the first row. The determinant can be written as:
[tex]\[
\text{det} =
1 \cdot \left|\begin{array}{cc}
5 & 0 \\
1 & 2
\end{array}\right|
- (-1) \cdot \left|\begin{array}{cc}
2 & 0 \\
-3 & 2
\end{array}\right|
+ 3 \cdot \left|\begin{array}{cc}
2 & 5 \\
-3 & 1
\end{array}\right|
\][/tex]
Now we need to calculate the determinants of the 2x2 matrices:
1. Calculate [tex]\(\left|\begin{array}{cc} 5 & 0 \\ 1 & 2 \end{array}\right|\)[/tex]:
[tex]\[
\left|\begin{array}{cc} 5 & 0 \\ 1 & 2 \end{array}\right| = (5 \cdot 2 - 0 \cdot 1) = 10
\][/tex]
2. Calculate [tex]\(\left|\begin{array}{cc} 2 & 0 \\ -3 & 2 \end{array}\right|\)[/tex]:
[tex]\[
\left|\begin{array}{cc} 2 & 0 \\ -3 & 2 \end{array}\right| = (2 \cdot 2 - 0 \cdot (-3)) = 4
\][/tex]
3. Calculate [tex]\(\left|\begin{array}{cc} 2 & 5 \\ -3 & 1 \end{array}\right|\)[/tex]:
[tex]\[
\left|\begin{array}{cc} 2 & 5 \\ -3 & 1 \end{array}\right| = (2 \cdot 1 - 5 \cdot (-3)) = 2 + 15 = 17
\][/tex]
Substituting these values back, we get:
[tex]\[
\text{det} = 1 \cdot 10 - (-1) \cdot 4 + 3 \cdot 17
\][/tex]
[tex]\[
= 10 + 4 + 51
\][/tex]
[tex]\[
= 65
\][/tex]
So, the value of the determinant is [tex]\(65\)[/tex].
