If `A+B=[(1,0,2),(2,2,2),(1,1,2)]` and `A-B=[(1,4,4),(4,2,0),(-1,1,2)]` then `A=[(1,2,3),(3,2,1),(0,0,2)]` and `B=[(0,-2,-1),(-1,0,1),(1,1,0)]`

Answer

To solve the problem, we need to find the matrices A and B given the equations:

1. \( A + B = \begin{pmatrix} 1 & 0 & 2 \\ 2 & 2 & 2 \\ 1 & 1 & 2 \end{pmatrix} \)
2. \( A - B = \begin{pmatrix} 1 & 4 & 4 \\ 4 & 2 & 0 \\ -1 & 1 & 2 \end{pmatrix} \)

### Step 1: Add the two equations

We will add the two equations together:

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

This simplifies to:

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

Calculating each element:

\[ 2A = \begin{pmatrix} 2 & 4 & 6 \\ 6 & 4 & 2 \\ 0 & 2 & 4 \end{pmatrix} \]

### Step 2: Divide by 2 to find A

Now, we divide each element by 2 to find matrix A:

\[ A = \frac{1}{2} \begin{pmatrix} 2 & 4 & 6 \\ 6 & 4 & 2 \\ 0 & 2 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix} \]

### Step 3: Substitute A back to find B

Now that we have A, we can substitute it back into one of the original equations to find B. We'll use the first equation:

\[ B = (A + B) - A \]

Substituting the values:

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

Calculating B:

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

### Final Matrices

Thus, we have:

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