To find the value of the determinant of the given [tex]\(3 \times 3\)[/tex] matrix, we can use the following formula for the determinant of a [tex]\(3 \times 3\)[/tex] matrix [tex]\(A\)[/tex]:
[tex]\[
\left| A \right| = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
where [tex]\(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\)[/tex].
For the given matrix:
[tex]\[
\left|\begin{array}{rrr}
2 & 1 & -2 \\
1 & -5 & 4 \\
1 & 5 & 5
\end{array}\right|
\][/tex]
We have:
- [tex]\(a = 2\)[/tex], [tex]\(b = 1\)[/tex], [tex]\(c = -2\)[/tex]
- [tex]\(d = 1\)[/tex], [tex]\(e = -5\)[/tex], [tex]\(f = 4\)[/tex]
- [tex]\(g = 1\)[/tex], [tex]\(h = 5\)[/tex], [tex]\(i = 5\)[/tex]
Using the formula, we plug in these values:
[tex]\[
|A| = 2((-5 \cdot 5) - (4 \cdot 5)) - 1((1 \cdot 5) - (4 \cdot 1)) + (-2)((1 \cdot 5) - (-5 \cdot 1))
\][/tex]
[tex]\[
|A| = 2((-25) - 20) - 1(5 - 4) + (-2)(5 + 5)
\][/tex]
[tex]\[
|A| = 2(-45) -1(1) + (-2)(10)
\][/tex]
[tex]\[
|A| = 2(-45) - 1 + (-2 \cdot 10)
\][/tex]
[tex]\[
|A| = -90 - 1 - 20
\][/tex]
[tex]\[
|A| = -111
\][/tex]
So, the determinant of the given matrix is [tex]\(-111\)[/tex].
