If `A=[{:(,1,3),(,3,4):}], B=[{:(,-2,1),(,-3,2):}] and A^2-5B^2=5C`. find matrix C where C is a 2 by 2 matrix.

Answer

To solve the problem, we need to find the matrix \( C \) given the matrices \( A \) and \( B \), and the equation \( A^2 - 5B^2 = 5C \).

### Step 1: Define the matrices \( A \) and \( B \)

Given:
\[ A = \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} -2 & 1 \\ -3 & 2 \end{pmatrix} \]

### Step 2: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:
\[ A^2 = A \times A = \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix} \times \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 1 \cdot 1 + 3 \cdot 3 = 1 + 9 = 10 \)
- First row, second column: \( 1 \cdot 3 + 3 \cdot 4 = 3 + 12 = 15 \)
- Second row, first column: \( 3 \cdot 1 + 4 \cdot 3 = 3 + 12 = 15 \)
- Second row, second column: \( 3 \cdot 3 + 4 \cdot 4 = 9 + 16 = 25 \)

Thus,
\[ A^2 = \begin{pmatrix} 10 & 15 \\ 15 & 25 \end{pmatrix} \]

### Step 3: Calculate \( B^2 \)

Now, we calculate \( B^2 \):
\[ B^2 = B \times B = \begin{pmatrix} -2 & 1 \\ -3 & 2 \end{pmatrix} \times \begin{pmatrix} -2 & 1 \\ -3 & 2 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( -2 \cdot -2 + 1 \cdot -3 = 4 - 3 = 1 \)
- First row, second column: \( -2 \cdot 1 + 1 \cdot 2 = -2 + 2 = 0 \)
- Second row, first column: \( -3 \cdot -2 + 2 \cdot -3 = 6 - 6 = 0 \)
- Second row, second column: \( -3 \cdot 1 + 2 \cdot 2 = -3 + 4 = 1 \)

Thus,
\[ B^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Calculate \( 5B^2 \)

Now we multiply \( B^2 \) by 5:
\[ 5B^2 = 5 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

### Step 5: Calculate \( A^2 - 5B^2 \)

Now we subtract \( 5B^2 \) from \( A^2 \):
\[ A^2 - 5B^2 = \begin{pmatrix} 10 & 15 \\ 15 & 25 \end{pmatrix} - \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 10 - 5 = 5 \)
- First row, second column: \( 15 - 0 = 15 \)
- Second row, first column: \( 15 - 0 = 15 \)
- Second row, second column: \( 25 - 5 = 20 \)

Thus,
\[ A^2 - 5B^2 = \begin{pmatrix} 5 & 15 \\ 15 & 20 \end{pmatrix} \]

### Step 6: Solve for \( C \)

According to the equation \( A^2 - 5B^2 = 5C \), we can express \( C \) as:
\[ 5C = A^2 - 5B^2 = \begin{pmatrix} 5 & 15 \\ 15 & 20 \end{pmatrix} \]

To find \( C \), we divide each element by 5:
\[ C = \frac{1}{5} \begin{pmatrix} 5 & 15 \\ 15 & 20 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix} \]

### Final Answer

Thus, the matrix \( C \) is:
\[ C = \begin{pmatrix} 1 & 3 \\ 3 & 4 \end{pmatrix} \]