\begin{tabular}{l} 2) Find \( b \) * \\ Given that \( A=\left(\begin{array}{cc}2 & -3 \\ 4 & 5\end{array}\right), B=\left(\begin{array}{ll}7 & 6 \\ 8 & 1\end{array}\right), C=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \) and \( \frac{1}{2} A-\frac{1}{4} B=C \), \\ calculate the values of \( a, b, c \) and \( d \). \\ Your answer \\ \hline\end{tabular}

To find the values of \( a, b, c, \) and \( d \) in the matrix \( C \), we start with the equation given: \[ \frac{1}{2} A - \frac{1}{4} B = C \] First, we need to calculate \( \frac{1}{2} A \) and \( \frac{1}{4} B \). ### Step 1: Calculate \( \frac{1}{2} A \) Given: \[ A = \begin{pmatrix} 2 & -3 \\ 4 & 5 \end{pmatrix} \] Calculating \( \frac{1}{2} A \): \[ \frac{1}{2} A = \frac{1}{2} \begin{pmatrix} 2 & -3 \\ 4 & 5 \end{pmatrix} = \begin{pmatrix} 1 & -\frac{3}{2} \\ 2 & \frac{5}{2} \end{pmatrix} \] ### Step 2: Calculate \( \frac{1}{4} B \) Given: \[ B = \begin{pmatrix} 7 & 6 \\ 8 & 1 \end{pmatrix} \] Calculating \( \frac{1}{4} B \): \[ \frac{1}{4} B = \frac{1}{4} \begin{pmatrix} 7 & 6 \\ 8 & 1 \end{pmatrix} = \begin{pmatrix} \frac{7}{4} & \frac{3}{2} \\ 2 & \frac{1}{4} \end{pmatrix} \] ### Step 3: Calculate \( \frac{1}{2} A - \frac{1}{4} B \) Now we subtract \( \frac{1}{4} B \) from \( \frac{1}{2} A \): \[ C = \frac{1}{2} A - \frac{1}{4} B = \begin{pmatrix} 1 & -\frac{3}{2} \\ 2 & \frac{5}{2} \end{pmatrix} - \begin{pmatrix} \frac{7}{4} & \frac{3}{2} \\ 2 & \frac{1}{4} \end{pmatrix} \] Calculating each element of \( C \): 1. For the element at (1,1): \[ 1 - \frac{7}{4} = \frac{4}{4} - \frac{7}{4} = -\frac{3}{4} \] 2. For the element at (1,2): \[ -\frac{3}{2} - \frac{3}{2} = -3 \] 3. For the element at (2,1): \[ 2 - 2 = 0 \] 4. For the element at (2,2): \[ \frac{5}{2} - \frac{1}{4} = \frac{10}{4} - \frac{1}{4} = \frac{9}{4} \] ### Step 4: Write the resulting matrix \( C \) Thus, we have: \[ C = \begin{pmatrix} -\frac{3}{4} & -3 \\ 0 & \frac{9}{4} \end{pmatrix} \] ### Step 5: Identify \( a, b, c, d \) From the matrix \( C \): - \( a = -\frac{3}{4} \) - \( b = -3 \) - \( c = 0 \) - \( d = \frac{9}{4} \) ### Final Answer The values are: \[ a = -\frac{3}{4}, \quad b = -3, \quad c = 0, \quad d = \frac{9}{4} \]