To find the determinant of the matrix [tex]\( \left|\begin{array}{lll}x^2 & x & 1 \\ x^3 & x^2 & x \\ x^4 & x^3 & x^2\end{array}\right| \)[/tex], we can proceed with the following steps:
1. Set Up the Matrix:
[tex]\[
A = \begin{pmatrix}
x^2 & x & 1 \\
x^3 & x^2 & x \\
x^4 & x^3 & x^2
\end{pmatrix}
\][/tex]
2. Calculate the Determinant:
The determinant of a [tex]\(3 \times 3\)[/tex] matrix [tex]\( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \)[/tex] is given by:
[tex]\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
3. Apply the Formula:
Let [tex]\( a = x^2 \)[/tex], [tex]\( b = x \)[/tex], [tex]\( c = 1 \)[/tex], [tex]\( d = x^3 \)[/tex], [tex]\( e = x^2 \)[/tex], [tex]\( f = x \)[/tex], [tex]\( g = x^4 \)[/tex], [tex]\( h = x^3 \)[/tex], and [tex]\( i = x^2 \)[/tex].
[tex]\[
\det(A) = x^2 \left( x^2 \cdot x^2 - x \cdot x^3 \right) - x \left( x^3 \cdot x^2 - x \cdot x^4 \right) + 1 \left( x^3 \cdot x^3 - x^2 \cdot x^4 \right)
\][/tex]
4. Simplify Each Term:
- Calculate [tex]\( ei - fh \)[/tex]:
[tex]\[
ei = x^2 \cdot x^2 = x^4
\][/tex]
[tex]\[
fh = x \cdot x^3 = x^4
\][/tex]
[tex]\[
ei - fh = x^4 - x^4 = 0
\][/tex]
- Calculate [tex]\( di - fg \)[/tex]:
[tex]\[
di = x^3 \cdot x^2 = x^5
\][/tex]
[tex]\[
fg = x \cdot x^4 = x^5
\][/tex]
[tex]\[
di - fg = x^5 - x^5 = 0
\][/tex]
- Calculate [tex]\( dh - eg \)[/tex]:
[tex]\[
dh = x^3 \cdot x^3 = x^6
\][/tex]
[tex]\[
eg = x^2 \cdot x^4 = x^6
\][/tex]
[tex]\[
dh - eg = x^6 - x^6 = 0
\][/tex]
5. Combine the Results:
[tex]\[
\det(A) = x^2 \cdot 0 - x \cdot 0 + 1 \cdot 0 = 0 - 0 + 0 = 0
\][/tex]
Therefore, the value of the determinant of the given matrix is [tex]\(\boxed{0}\)[/tex].
