If `A=[(2,-1),(4,2)]` and `B=[(2,3),(1,2)]`find `2A+3B`

Answer

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

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

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

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

Calculating the elements:

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

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

Next, we calculate \( 3B \) by multiplying each element of matrix \( B \) by 3.

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

Calculating the elements:

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

### Step 3: Add \( 2A \) and \( 3B \)

Now we will add the matrices \( 2A \) and \( 3B \):

\[ 2A + 3B = \begin{pmatrix} 4 & -2 \\ 8 & 4 \end{pmatrix} + \begin{pmatrix} 6 & 9 \\ 3 & 6 \end{pmatrix} \]

To perform the addition, we add the corresponding elements:

\[ = \begin{pmatrix} 4 + 6 & -2 + 9 \\ 8 + 3 & 4 + 6 \end{pmatrix} \]

Calculating the elements:

\[ = \begin{pmatrix} 10 & 7 \\ 11 & 10 \end{pmatrix} \]

### Final Answer

Thus, \( 2A + 3B = \begin{pmatrix} 10 & 7 \\ 11 & 10 \end{pmatrix} \).

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