To solve the matrix inequality [tex]\(\left[\begin{array}{ccc}
2 - x & x & 1 \\
2 + x & -x + 2 & -3 \\
x & 3 & 0
\end{array}\right] \leq 0\)[/tex], we need to find the values of [tex]\(x\)[/tex] for which every element in the matrix is less than or equal to 0.
Let's analyze each element individually:
1. [tex]\(2 - x \leq 0\)[/tex]:
[tex]\[ 2 - x \leq 0 \][/tex]
[tex]\[ -x \leq -2 \][/tex]
[tex]\[ x \geq 2 \][/tex]
2. [tex]\(x \leq 0\)[/tex]:
[tex]\[ x \leq 0 \][/tex]
3. [tex]\(1 \leq 0\)[/tex]:
[tex]\[ 1 \leq 0 \][/tex]
This inequality is not true for any [tex]\(x\)[/tex]. It means that this particular matrix cannot satisfy the given inequality for all elements because [tex]\(1 \leq 0\)[/tex] is never true no matter what value [tex]\(x\)[/tex] takes.
Since one of the conditions cannot be met (specifically, [tex]\(1 \leq 0\)[/tex]), it implies that there are no values of [tex]\(x\)[/tex] that can make every single element in this matrix less than or equal to 0.
Therefore, there are no solutions for [tex]\(x\)[/tex] that satisfy the entire matrix inequality.
