If `A=[(2,3,4),(1,2,3),(-1,1,2)],B=[(1,3,0),(-1,2,1),(0,0,2)]`, then AB+BA=………….

Answer

To solve the problem, we need to calculate the matrices \( AB \) and \( BA \) and then add them together. Given the matrices:

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

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

To find \( AB \), we multiply matrix \( A \) by matrix \( B \).

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

Calculating each element of the resulting matrix \( AB \):

- First row, first column:
\[ 2 \cdot 1 + 3 \cdot (-1) + 4 \cdot 0 = 2 - 3 + 0 = -1 \]

- First row, second column:
\[ 2 \cdot 3 + 3 \cdot 2 + 4 \cdot 0 = 6 + 6 + 0 = 12 \]

- First row, third column:
\[ 2 \cdot 0 + 3 \cdot 1 + 4 \cdot 2 = 0 + 3 + 8 = 11 \]

- Second row, first column:
\[ 1 \cdot 1 + 2 \cdot (-1) + 3 \cdot 0 = 1 - 2 + 0 = -1 \]

- Second row, second column:
\[ 1 \cdot 3 + 2 \cdot 2 + 3 \cdot 0 = 3 + 4 + 0 = 7 \]

- Second row, third column:
\[ 1 \cdot 0 + 2 \cdot 1 + 3 \cdot 2 = 0 + 2 + 6 = 8 \]

- Third row, first column:
\[ -1 \cdot 1 + 1 \cdot (-1) + 2 \cdot 0 = -1 - 1 + 0 = -2 \]

- Third row, second column:
\[ -1 \cdot 3 + 1 \cdot 2 + 2 \cdot 0 = -3 + 2 + 0 = -1 \]

- Third row, third column:
\[ -1 \cdot 0 + 1 \cdot 1 + 2 \cdot 2 = 0 + 1 + 4 = 5 \]

Thus, the matrix \( AB \) is:
\[ AB = \begin{pmatrix} -1 & 12 & 11 \\ -1 & 7 & 8 \\ -2 & -1 & 5 \end{pmatrix} \]

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

Now we calculate \( BA \) by multiplying matrix \( B \) by matrix \( A \).

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

Calculating each element of the resulting matrix \( BA \):

- First row, first column:
\[ 1 \cdot 2 + 3 \cdot 1 + 0 \cdot (-1) = 2 + 3 + 0 = 5 \]

- First row, second column:
\[ 1 \cdot 3 + 3 \cdot 2 + 0 \cdot 1 = 3 + 6 + 0 = 9 \]

- First row, third column:
\[ 1 \cdot 4 + 3 \cdot 3 + 0 \cdot 2 = 4 + 9 + 0 = 13 \]

- Second row, first column:
\[ -1 \cdot 2 + 2 \cdot 1 + 1 \cdot (-1) = -2 + 2 - 1 = -1 \]

- Second row, second column:
\[ -1 \cdot 3 + 2 \cdot 2 + 1 \cdot 1 = -3 + 4 + 1 = 2 \]

- Second row, third column:
\[ -1 \cdot 4 + 2 \cdot 3 + 1 \cdot 2 = -4 + 6 + 2 = 4 \]

- Third row, first column:
\[ 0 \cdot 2 + 0 \cdot 1 + 2 \cdot (-1) = 0 + 0 - 2 = -2 \]

- Third row, second column:
\[ 0 \cdot 3 + 0 \cdot 2 + 2 \cdot 1 = 0 + 0 + 2 = 2 \]

- Third row, third column:
\[ 0 \cdot 4 + 0 \cdot 3 + 2 \cdot 2 = 0 + 0 + 4 = 4 \]

Thus, the matrix \( BA \) is:
\[ BA = \begin{pmatrix} 5 & 9 & 13 \\ -1 & 2 & 4 \\ -2 & 2 & 4 \end{pmatrix} \]

### Step 3: Calculate \( AB + BA \)

Now we add the matrices \( AB \) and \( BA \):

\[ AB + BA = \begin{pmatrix} -1 & 12 & 11 \\ -1 & 7 & 8 \\ -2 & -1 & 5 \end{pmatrix} + \begin{pmatrix} 5 & 9 & 13 \\ -1 & 2 & 4 \\ -2 & 2 & 4 \end{pmatrix} \]

Adding the corresponding elements:

- First row:
\[ -1 + 5 = 4, \quad 12 + 9 = 21, \quad 11 + 13 = 24 \]

- Second row:
\[ -1 + (-1) = -2, \quad 7 + 2 = 9, \quad 8 + 4 = 12 \]

- Third row:
\[ -2 + (-2) = -4, \quad -1 + 2 = 1, \quad 5 + 4 = 9 \]

Thus, the final result is:
\[ AB + BA = \begin{pmatrix} 4 & 21 & 24 \\ -2 & 9 & 12 \\ -4 & 1 & 9 \end{pmatrix} \]

### Final Answer:
\[ AB + BA = \begin{pmatrix} 4 & 21 & 24 \\ -2 & 9 & 12 \\ -4 & 1 & 9 \end{pmatrix} \]