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Let's begin by solving part **a** using the repeated column method to evaluate the determinant of matrix \( A \).

The matrix \( A \) is given as:

\[
A = \begin{pmatrix}
2 & 2 & x \\
2 & x & x \\
x & x & x
\end{pmatrix}
\]

### Part a: Rewrite the first two columns and evaluate the determinant

The **repeated column method** involves repeating the first two columns of the matrix beside the third column to form an extended matrix. This helps with applying the rule for calculating the determinant of a 3x3 matrix.

We extend the matrix as follows:

\[
\begin{pmatrix}
2 & 2 & x & 2 & 2 \\
2 & x & x & 2 & x \\
x & x & x & x & x
\end{pmatrix}
\]

To find the determinant of the original matrix, we use the following formula for a 3x3 matrix:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Where:

\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
=
\begin{pmatrix}
2 & 2 & x \\
2 & x & x \\
x & x & x
\end{pmatrix}
\]

Now, using the determinant formula:

- \( a = 2, b = 2, c = x \)
- \( d = 2, e = x, f = x \)
- \( g = x, h = x, i = x \)

We calculate each term:

1. \( a(ei - fh) = 2(x \cdot x - x \cdot x) = 2(0) = 0 \)
2. \( -b(di - fg) = -2(2 \cdot x - x \cdot x) = -2(2x - x^2) = -4x + 2x^2 \)
3. \( c(dh - eg) = x(2 \cdot x - x \cdot x) = x(2x - x^2) = 2x^2 - x^3 \)

So, the determinant is:

\[
\text{det}(A) = 0 - 4x + 2x^2 + 2x^2 - x^3
\]
\[
\text{det}(A) = -x^3 + 4x^2 - 4x
\]

### Part b: Determine \( x \) so that \( A \) will not have an inverse

A matrix does not have an inverse if its determinant is **zero**. Thus, we solve for \( x \) such that:

\[
-x^3 + 4x^2 - 4x = 0
\]

Factoring the equation:

\[
-x(x^2 - 4x + 4) = 0
\]
\[
-x(x - 2)^2 = 0
\]

This gives two solutions:

1. \( x = 0 \)
2. \( x = 2 \) (with multiplicity 2)

Thus, \( A \) will **not** have an inverse when \( x = 0 \) or \( x = 2 \).

---

Would you like more details or have any questions? Here are some related questions you could explore:

1. What is the general condition for a matrix to not have an inverse?
2. How can the determinant help identify singular matrices?
3. How would the solution change if the matrix had different entries?
4. What happens to the determinant if one of the rows or columns is entirely zero?
5. Can this method be applied to larger matrices (e.g., 4x4)?

**Tip:** When factoring cubic polynomials, look for common factors first to simplify the equation.

Given the matrix A = [[2, 2, x], [2, x, x], [x, x, x]], rewrite the first two columns (repeated columns) to evaluate the determinant of A. Determine x so that A will not have an inverse.

Learn how to evaluate the determinant of a 3x3 matrix with repeated columns and determine the conditions under which the matrix will not have an inverse. This problem covers key linear algebra concepts, including matrix inversion and factoring cubic polynomials. Suitable for students in Grades 10-12.

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