Alright! Let's calculate the determinant of the given 3x3 matrix step-by-step. The matrix is:
[tex]\[
\begin{pmatrix}
1-2 & 4 & 2 \\
1 & 1+1 & -5 \\
0 & 1 & 1-14
\end{pmatrix}
\][/tex]
First, we simplify each entry:
[tex]\[
\begin{pmatrix}
-1 & 4 & 2 \\
1 & 2 & -5 \\
0 & 1 & -13
\end{pmatrix}
\][/tex]
Now, to find the determinant, we can use the formula for the determinant of a 3x3 matrix:
[tex]\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
where the matrix [tex]\( A \)[/tex] is:
[tex]\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\][/tex]
For our matrix:
[tex]\[
\begin{pmatrix}
-1 & 4 & 2 \\
1 & 2 & -5 \\
0 & 1 & -13
\end{pmatrix}
\][/tex]
we can identify the variables as follows:
- [tex]\(a = -1\)[/tex]
- [tex]\(b = 4\)[/tex]
- [tex]\(c = 2\)[/tex]
- [tex]\(d = 1\)[/tex]
- [tex]\(e = 2\)[/tex]
- [tex]\(f = -5\)[/tex]
- [tex]\(g = 0\)[/tex]
- [tex]\(h = 1\)[/tex]
- [tex]\(i = -13\)[/tex]
Now we substitute these into the determinant formula:
[tex]\[
\text{det}(A) = -1 \cdot (2 \cdot -13 - (-5) \cdot 1) - 4 \cdot (1 \cdot -13 - (-5) \cdot 0) + 2 \cdot (1 \cdot 1 - 2 \cdot 0)
\][/tex]
We simplify inside the parentheses first:
[tex]\[
2 \cdot -13 - (-5) \cdot 1 = -26 + 5 = -21
\][/tex]
[tex]\[
1 \cdot -13 - (-5) \cdot 0 = -13
\][/tex]
[tex]\[
1 \cdot 1 - 2 \cdot 0 = 1
\][/tex]
Now substitute these values back into the equation:
[tex]\[
\text{det}(A) = -1 \cdot (-21) - 4 \cdot (-13) + 2 \cdot 1
\][/tex]
Now, calculate each term:
[tex]\[
-1 \cdot (-21) = 21
\][/tex]
[tex]\[
-4 \cdot (-13) = 52
\][/tex]
[tex]\[
2 \cdot 1 = 2
\][/tex]
Add these results together:
[tex]\[
21 + 52 + 2 = 75
\][/tex]
Hence, the determinant of the matrix is:
[tex]\[
\boxed{75}
\][/tex]
