To find the inverses of the given matrices $A$, $B$, and $C$, we need to check if each matrix is invertible. A matrix is invertible if its determinant is non-zero. If the determinant is zero, the matrix is singular, and no inverse exists.

Matrix $A$:

$A = \begin{pmatrix} 3 & 1 \\ 3 & 2 \end{pmatrix}$ The determinant of $A$ is: $\text{det}(A) = (3)(2) - (1)(3) = 6 - 3 = 3$ Since the determinant of $A$ is 3 (non-zero), the matrix is invertible. The inverse of a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is given by: $A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ Applying this formula to matrix $A$: $A^{-1} = \frac{1}{3} \begin{pmatrix} 2 & -1 \\ -3 & 3 \end{pmatrix} = \begin{pmatrix} \frac{2}{3} & -\frac{1}{3} \\ -1 & 1 \end{pmatrix}$

Matrix $B$:

$B = \begin{pmatrix} 1 & 4 \\ -3 & 2 \end{pmatrix}$ The determinant of $B$ is: $\text{det}(B) = (1)(2) - (4)(-3) = 2 + 12 = 14$ Since the determinant of $B$ is 14 (non-zero), the matrix is invertible. Using the inverse formula: $B^{-1} = \frac{1}{14} \begin{pmatrix} 2 & -4 \\ 3 & 1 \end{pmatrix} = \begin{pmatrix} \frac{2}{14} & -\frac{4}{14} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix} = \begin{pmatrix} \frac{1}{7} & -\frac{2}{7} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix}$

Matrix $C$:

$C = \begin{pmatrix} 3 & 4 \\ -3 & -6 \end{pmatrix}$ The determinant of $C$ is: $\text{det}(C) = (3)(-6) - (4)(-3) = -18 + 12 = -6$ Since the determinant of $C$ is -6 (non-zero), the matrix is invertible. Using the inverse formula: $C^{-1} = \frac{1}{-6} \begin{pmatrix} -6 & -4 \\ 3 & 3 \end{pmatrix} = \begin{pmatrix} 1 & \frac{2}{3} \\ -\frac{1}{2} & -\frac{1}{2} \end{pmatrix}$

Summary:

• $A^{-1} = \begin{pmatrix} \frac{2}{3} & -\frac{1}{3} \\ -1 & 1 \end{pmatrix}$
• $B^{-1} = \begin{pmatrix} \frac{1}{7} & -\frac{2}{7} \\ \frac{3}{14} & \frac{1}{14} \end{pmatrix}$
• $C^{-1} = \begin{pmatrix} 1 & \frac{2}{3} \\ -\frac{1}{2} & -\frac{1}{2} \end{pmatrix}$

Would you like more details on the calculation steps or have any other questions?

Here are some follow-up questions for further exploration:

1. What is the significance of the determinant in determining the invertibility of a matrix?
2. How does matrix inversion apply to solving systems of linear equations?
3. Can we use a different method, like Gaussian elimination, to find the inverse of a matrix?
4. How does the inverse of a matrix relate to its transpose and adjugate?
5. What is the computational complexity of finding the inverse of a matrix in larger systems?

Tip: Always check the determinant before attempting to find the inverse of a matrix, as it will save time if the matrix is singular.