To determine the value of [tex]\(x\)[/tex] that will make the matrix [tex]\(\left[\begin{array}{cc} x-2 & 3 \\ 1 & 3 \end{array}\right]\)[/tex] singular, we need to find when the determinant of the matrix is zero.
A matrix is singular if and only if its determinant is zero. We can calculate the determinant of a [tex]\(2 \times 2\)[/tex] matrix [tex]\(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\)[/tex] using the formula:
[tex]\[
\text{det} = ad - bc
\][/tex]
For the given matrix [tex]\(\left[\begin{array}{cc} x-2 & 3 \\ 1 & 3 \end{array}\right]\)[/tex]:
1. Identify the elements [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] in the matrix. In our case:
[tex]\[
a = x - 2, \quad b = 3, \quad c = 1, \quad d = 3
\][/tex]
2. Apply the determinant formula:
[tex]\[
\text{det} = (x - 2) \cdot 3 - 3 \cdot 1
\][/tex]
3. Simplify the expression:
[tex]\[
\text{det} = 3(x - 2) - 3
\][/tex]
[tex]\[
\text{det} = 3x - 6 - 3
\][/tex]
[tex]\[
\text{det} = 3x - 9
\][/tex]
4. Set the determinant equal to zero to find the value of [tex]\(x\)[/tex]:
[tex]\[
3x - 9 = 0
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
3x = 9
\][/tex]
[tex]\[
x = 3
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that makes the matrix singular is:
[tex]\[
x = 3
\][/tex]
