Sure, let's solve the given determinant equation step-by-step to find the value(s) of [tex]\( x \)[/tex].
We are given the following matrix and need to find its determinant:
[tex]\[
\left| \begin{array}{ccc} x-1 & 1 & 1 \\ -1 & x & -2 \\ 1 & 0 & 2 \end{array} \right| = 0
\][/tex]
Let's calculate the determinant of this 3x3 matrix. The determinant of a 3x3 matrix:
[tex]\[
\left| \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right|
\][/tex]
is given by:
[tex]\[
a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
In our case, the elements are:
[tex]\( a = x-1 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = 1 \)[/tex] \\
[tex]\( d = -1 \)[/tex], [tex]\( e = x \)[/tex], [tex]\( f = -2 \)[/tex] \\
[tex]\( g = 1 \)[/tex], [tex]\( h = 0 \)[/tex], [tex]\( i = 2 \)[/tex]
Now, let's use the determinant formula:
[tex]\[
\left| \begin{array}{ccc} x-1 & 1 & 1 \\ -1 & x & -2 \\ 1 & 0 & 2 \end{array} \right| = (x-1) \cdot (x \cdot 2 - (-2) \cdot 0) - 1 \cdot (-1 \cdot 2 - (-2) \cdot 1) + 1 \cdot (-1 \cdot 0 - x \cdot 1)
\][/tex]
Let's simplify each term:
- The first term:
[tex]\[
(x-1) \cdot (2x - 0) = (x-1) \cdot 2x = 2x^2 - 2x
\][/tex]
- The second term:
[tex]\[
1 \cdot (-2 + 2) = 1 \cdot 0 = 0
\][/tex]
- The third term:
[tex]\[
1 \cdot (0 - x) = 1 \cdot (-x) = -x
\][/tex]
Combining these, we get the determinant:
[tex]\[
2x^2 - 2x + 0 - x = 2x^2 - 3x
\][/tex]
So, the determinant simplifies to:
[tex]\[
2x^2 - 3x
\][/tex]
We need to find the value(s) of [tex]\( x \)[/tex] for which the determinant is zero. Therefore, we set up the equation:
[tex]\[
2x^2 - 3x = 0
\][/tex]
Factor out [tex]\( x \)[/tex]:
[tex]\[
x(2x - 3) = 0
\][/tex]
Now, solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0 \quad \text{or} \quad 2x - 3 = 0
\][/tex]
Solving [tex]\( 2x - 3 = 0 \)[/tex]:
[tex]\[
2x = 3
\][/tex]
[tex]\[
x = \frac{3}{2}
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] that satisfy the equation are:
[tex]\[
x = 0 \quad \text{or} \quad x = \frac{3}{2}
\][/tex]
