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.