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To determine for which value of \( x \) the inverse of the matrix \(\begin{pmatrix} 4 & -3 \\ x & 3 \end{pmatrix}\) cannot be defined, we need to find the determinant of the matrix and set it equal to zero. The inverse of a matrix exists if and only if the determinant of the matrix is non-zero.

The determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by \( ad - bc \).

For the given matrix \(\begin{pmatrix} 4 & -3 \\ x & 3 \end{pmatrix}\):

\[
\text{Determinant} = (4 \cdot 3) - (-3 \cdot x) = 12 + 3x
\]

We need to set the determinant to zero to find the value of \( x \) for which the inverse cannot be defined:

\[
12 + 3x = 0
\]

Solving for \( x \):

\[
3x = -12
\]

\[
x = -4
\]

Therefore, the inverse of the matrix \(\begin{pmatrix} 4 & -3 \\ x & 3 \end{pmatrix}\) cannot be defined when \( x = -4 \).

If you have any further questions or need more details, feel free to ask!

Here are 5 questions you might want to ask next:
1. How do you find the inverse of a \(2 \times 2\) matrix?
2. What is the significance of the determinant in matrix algebra?
3. Can you explain the properties of determinants?
4. How are eigenvalues related to the determinant of a matrix?
5. What are some applications of matrix inverses in real-world problems?

**Tip:** When dealing with matrices, always check the determinant before attempting to find the inverse. If the determinant is zero, the matrix does not have an inverse.

Learn how to determine the value of x for which the inverse of the matrix cannot be defined. This problem involves matrix algebra, specifically focusing on the determinant of a 2x2 matrix and its relationship to matrix invertibility. Suitable for high school students studying algebra and matrix theory.

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