Given that A = [2 4; 3 6] and B = [11 3; 4 1] find C such that BC = A. 2. Find the inverse of the matrix [2 5; 3 4]. Hence or otherwise solve the equations 2x + 5y = 9 and 3x + 4y = 6.

here are the solutions to the matrix problems:

1. Given that $A = \begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix}$ and $B = \begin{pmatrix} 11 & 3 \\ 4 & 1 \end{pmatrix}$, find $C$ such that $B \cdot C = A$.

Step 1: To find $C$, we need to multiply both sides of the equation $B \cdot C = A$ by the inverse of $B$ ($B^{-1}$) from the left. This gives $C = B^{-1} \cdot A$. First, calculate the determinant of $B$.

$$\det(B) = (11)(1) - (3)(4) = 11 - 12 = -1$$

Step 2: Calculate the inverse of $B$.

$$B^{-1} = \frac{1}{\det(B)} \begin{pmatrix} 1 & -3 \\ -4 & 11 \end{pmatrix} = \frac{1}{-1} \begin{pmatrix} 1 & -3 \\ -4 & 11 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ 4 & -11 \end{pmatrix}$$

Step 3: Multiply $B^{-1}$ by $A$ to find $C$.

$$C = B^{-1} \cdot A = \begin{pmatrix} -1 & 3 \\ 4 & -11 \end{pmatrix} \begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix}$$ $$C = \begin{pmatrix} (-1)(2) + (3)(3) & (-1)(4) + (3)(6) \\ (4)(2) + (-11)(3) & (4)(4) + (-11)(6) \end{pmatrix}$$ $$C = \begin{pmatrix} -2 + 9 & -4 + 18 \\ 8 - 33 & 16 - 66 \end{pmatrix}$$ $$C = \begin{pmatrix} 7 & 14 \\ -25 & -50 \end{pmatrix}$$

The matrix $C$ is $\boxed{\begin{pmatrix} 7 & 14 \\ -25 & -50 \end{pmatrix}}$.

2. Find the inverse of the matrix $\begin{pmatrix} 2 & 5 \\ 3 & 4 \end{pmatrix}$. Hence or otherwise solve the equations $2x + 5y = 9$ and $3x + 4y = 6$.