If `A'=[(-2,3),(1,3)] and B=[(-1,0),(1,2)],` find `[A+2B]'.`

Explanation

To solve the problem of finding \([A + 2B]'\) where \(A' = \begin{pmatrix} -2 & 3 \\ 1 & 3 \end{pmatrix}\) and \(B = \begin{pmatrix} -1 & 0 \\ 1 & 2 \end{pmatrix}\), we will follow these steps:

### Step 1: Find matrix \(A\)
Since \(A'\) is given, we can find \(A\) by taking the transpose of \(A'\):
\[ A = (A')' = \begin{pmatrix} -2 & 1 \\ 3 & 3 \end{pmatrix} \]

### Step 2: Calculate \(2B\)
Next, we need to calculate \(2B\):
\[ B = \begin{pmatrix} -1 & 0 \\ 1 & 2 \end{pmatrix} \]
\[ 2B = 2 \times \begin{pmatrix} -1 & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} -2 & 0 \\ 2 & 4 \end{pmatrix} \]

### Step 3: Add \(A\) and \(2B\)
Now we will add \(A\) and \(2B\):
\[ A + 2B = \begin{pmatrix} -2 & 1 \\ 3 & 3 \end{pmatrix} + \begin{pmatrix} -2 & 0 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} -2 + (-2) & 1 + 0 \\ 3 + 2 & 3 + 4 \end{pmatrix} \]
\[ A + 2B = \begin{pmatrix} -4 & 1 \\ 5 & 7 \end{pmatrix} \]

### Step 4: Find the transpose of \(A + 2B\)
Finally, we need to find the transpose of the resulting matrix:
\[ [A + 2B]' = \begin{pmatrix} -4 & 1 \\ 5 & 7 \end{pmatrix}' = \begin{pmatrix} -4 & 5 \\ 1 & 7 \end{pmatrix} \]

### Final Answer
Thus, the final answer is:
\[ [A + 2B]' = \begin{pmatrix} -4 & 5 \\ 1 & 7 \end{pmatrix} \]