If `A=[{:(,3,7),(,2,4):}], B=[{:(,0,2),(,5,3):}] and C=[{:(,1,-5),(,-4,6):}]` Find AB-5C.

Answer

To solve the problem, we need to compute \( AB - 5C \) where:

\[ A = \begin{pmatrix} 3 & 7 \\ 2 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 2 \\ 5 & 3 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & -5 \\ -4 & 6 \end{pmatrix} \]

### Step 1: Calculate the product \( AB \)

To find \( AB \), we multiply matrix \( A \) by matrix \( B \).

\[ AB = \begin{pmatrix} 3 & 7 \\ 2 & 4 \end{pmatrix} \begin{pmatrix} 0 & 2 \\ 5 & 3 \end{pmatrix} \]

Calculating each element of the resulting matrix:

- **First row, first column**:
\[ 3 \cdot 0 + 7 \cdot 5 = 0 + 35 = 35 \]

- **First row, second column**:
\[ 3 \cdot 2 + 7 \cdot 3 = 6 + 21 = 27 \]

- **Second row, first column**:
\[ 2 \cdot 0 + 4 \cdot 5 = 0 + 20 = 20 \]

- **Second row, second column**:
\[ 2 \cdot 2 + 4 \cdot 3 = 4 + 12 = 16 \]

Thus, we have:
\[ AB = \begin{pmatrix} 35 & 27 \\ 20 & 16 \end{pmatrix} \]

### Step 2: Calculate \( 5C \)

Next, we need to multiply matrix \( C \) by 5.

\[ 5C = 5 \cdot \begin{pmatrix} 1 & -5 \\ -4 & 6 \end{pmatrix} = \begin{pmatrix} 5 \cdot 1 & 5 \cdot -5 \\ 5 \cdot -4 & 5 \cdot 6 \end{pmatrix} = \begin{pmatrix} 5 & -25 \\ -20 & 30 \end{pmatrix} \]

### Step 3: Calculate \( AB - 5C \)

Now we subtract \( 5C \) from \( AB \).

\[ AB - 5C = \begin{pmatrix} 35 & 27 \\ 20 & 16 \end{pmatrix} - \begin{pmatrix} 5 & -25 \\ -20 & 30 \end{pmatrix} \]

Calculating each element:

- **First row, first column**:
\[ 35 - 5 = 30 \]

- **First row, second column**:
\[ 27 - (-25) = 27 + 25 = 52 \]

- **Second row, first column**:
\[ 20 - (-20) = 20 + 20 = 40 \]

- **Second row, second column**:
\[ 16 - 30 = -14 \]

Thus, we have:
\[ AB - 5C = \begin{pmatrix} 30 & 52 \\ 40 & -14 \end{pmatrix} \]

### Final Answer

The final result is:
\[ AB - 5C = \begin{pmatrix} 30 & 52 \\ 40 & -14 \end{pmatrix} \]

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