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

Answer

To solve the problem, we need to find the expression \( AB + 2C - 4D \) given the matrices \( A \), \( B \), \( C \), and \( D \).

Given:
- \( A = \begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix} \)
- \( B = \begin{pmatrix} 6 \\ 1 \end{pmatrix} \)
- \( C = \begin{pmatrix} -4 \\ -5 \end{pmatrix} \)
- \( D = \begin{pmatrix} 2 \\ 2 \end{pmatrix} \)

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

To find \( AB \), we perform matrix multiplication:

\[ AB = \begin{pmatrix} 3 & -2 \\ -1 & 4 \end{pmatrix} \begin{pmatrix} 6 \\ 1 \end{pmatrix} \]

Calculating the elements:
- First row, first column: \( 3 \cdot 6 + (-2) \cdot 1 = 18 - 2 = 16 \)
- Second row, first column: \( -1 \cdot 6 + 4 \cdot 1 = -6 + 4 = -2 \)

Thus,

\[ AB = \begin{pmatrix} 16 \\ -2 \end{pmatrix} \]

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

Next, we calculate \( 2C \):

\[ 2C = 2 \begin{pmatrix} -4 \\ -5 \end{pmatrix} = \begin{pmatrix} 2 \cdot -4 \\ 2 \cdot -5 \end{pmatrix} = \begin{pmatrix} -8 \\ -10 \end{pmatrix} \]

### Step 3: Calculate \( 4D \)

Now, we calculate \( 4D \):

\[ 4D = 4 \begin{pmatrix} 2 \\ 2 \end{pmatrix} = \begin{pmatrix} 4 \cdot 2 \\ 4 \cdot 2 \end{pmatrix} = \begin{pmatrix} 8 \\ 8 \end{pmatrix} \]

### Step 4: Combine the results to find \( AB + 2C - 4D \)

Now we can substitute the results into the expression \( AB + 2C - 4D \):

\[ AB + 2C - 4D = \begin{pmatrix} 16 \\ -2 \end{pmatrix} + \begin{pmatrix} -8 \\ -10 \end{pmatrix} - \begin{pmatrix} 8 \\ 8 \end{pmatrix} \]

Calculating the addition and subtraction:
- First element: \( 16 - 8 - 8 = 0 \)
- Second element: \( -2 - 10 - 8 = -20 \)

Thus,

\[ AB + 2C - 4D = \begin{pmatrix} 0 \\ -20 \end{pmatrix} \]

### Final Answer:

The final result is:

\[ AB + 2C - 4D = \begin{pmatrix} 0 \\ -20 \end{pmatrix} \]

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