`if A=[{:(1,2,-5),(-3,4,6):}]and B[{:(-2,3,-4),(1,2,3):}]'` then find 2A+B.

Explanation

To solve the problem, we need to find \(2A + B\) where:

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

### Step 1: Calculate \(2A\)

To find \(2A\), we multiply each element of matrix \(A\) by 2.

\[ A = \begin{pmatrix} 1 & 2 & -5 \\ -3 & 4 & 6 \end{pmatrix} \]

Calculating \(2A\):

\[ 2A = 2 \cdot \begin{pmatrix} 1 & 2 & -5 \\ -3 & 4 & 6 \end{pmatrix} = \begin{pmatrix} 2 \cdot 1 & 2 \cdot 2 & 2 \cdot -5 \\ 2 \cdot -3 & 2 \cdot 4 & 2 \cdot 6 \end{pmatrix} \]

\[ = \begin{pmatrix} 2 & 4 & -10 \\ -6 & 8 & 12 \end{pmatrix} \]

### Step 2: Add \(2A\) and \(B\)

Now, we need to add \(2A\) to \(B\):

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

Now, we perform the addition:

\[ 2A + B = \begin{pmatrix} 2 & 4 & -10 \\ -6 & 8 & 12 \end{pmatrix} + \begin{pmatrix} -2 & 3 & -4 \\ 1 & 2 & 3 \end{pmatrix} \]

Adding the corresponding elements:

\[ = \begin{pmatrix} 2 + (-2) & 4 + 3 & -10 + (-4) \\ -6 + 1 & 8 + 2 & 12 + 3 \end{pmatrix} \]

Calculating each element:

1. \(2 + (-2) = 0\)
2. \(4 + 3 = 7\)
3. \(-10 + (-4) = -14\)
4. \(-6 + 1 = -5\)
5. \(8 + 2 = 10\)
6. \(12 + 3 = 15\)

So, we have:

\[ 2A + B = \begin{pmatrix} 0 & 7 & -14 \\ -5 & 10 & 15 \end{pmatrix} \]

### Final Answer

\[ 2A + B = \begin{pmatrix} 0 & 7 & -14 \\ -5 & 10 & 15 \end{pmatrix} \]

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