`if A=[{:(4,2),(-3,2),(1,3):}]and B=[{:(-1,3),(0,2),(2,-4):}],`then find 3A-4B.

Explanation

To find \( 3A - 4B \) where \( A = \begin{pmatrix} 4 & 2 \\ -3 & 2 \\ 1 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} -1 & 3 \\ 0 & 2 \\ 2 & -4 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( 3A \)

Multiply each element of matrix \( A \) by 3.

\[ 3A = 3 \times \begin{pmatrix} 4 & 2 \\ -3 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 3 \times 4 & 3 \times 2 \\ 3 \times -3 & 3 \times 2 \\ 3 \times 1 & 3 \times 3 \end{pmatrix} \]

Calculating each element:

\[ 3A = \begin{pmatrix} 12 & 6 \\ -9 & 6 \\ 3 & 9 \end{pmatrix} \]

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

Multiply each element of matrix \( B \) by 4.

\[ 4B = 4 \times \begin{pmatrix} -1 & 3 \\ 0 & 2 \\ 2 & -4 \end{pmatrix} = \begin{pmatrix} 4 \times -1 & 4 \times 3 \\ 4 \times 0 & 4 \times 2 \\ 4 \times 2 & 4 \times -4 \end{pmatrix} \]

Calculating each element:

\[ 4B = \begin{pmatrix} -4 & 12 \\ 0 & 8 \\ 8 & -16 \end{pmatrix} \]

### Step 3: Calculate \( 3A - 4B \)

Now we need to subtract \( 4B \) from \( 3A \).

\[ 3A - 4B = \begin{pmatrix} 12 & 6 \\ -9 & 6 \\ 3 & 9 \end{pmatrix} - \begin{pmatrix} -4 & 12 \\ 0 & 8 \\ 8 & -16 \end{pmatrix} \]

Subtracting the corresponding elements:

\[ = \begin{pmatrix} 12 - (-4) & 6 - 12 \\ -9 - 0 & 6 - 8 \\ 3 - 8 & 9 - (-16) \end{pmatrix} \]

Calculating each element:

\[ = \begin{pmatrix} 12 + 4 & 6 - 12 \\ -9 & 6 - 8 \\ 3 - 8 & 9 + 16 \end{pmatrix} \]

\[ = \begin{pmatrix} 16 & -6 \\ -9 & -2 \\ -5 & 25 \end{pmatrix} \]

### Final Answer

Thus, \( 3A - 4B = \begin{pmatrix} 16 & -6 \\ -9 & -2 \\ -5 & 25 \end{pmatrix} \).

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