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Question 3: Calculate $AC$, $BC$, and $(A + B)C$. Also verify that $(A + B)C = AC + BC$ for matrices $A$, $B$, and $C$.

Given matrices: $A = \begin{pmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \end{pmatrix}$ $B = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 2 & 0 \end{pmatrix}$ $C = \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 0 & -2 \end{pmatrix}$

Let's calculate $AC$, $BC$, and $(A + B)C$, and then verify the equation $(A + B)C = AC + BC$.

Step 1: Calculating $AC$

$AC = A \cdot C = \begin{pmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 0 & -2 \end{pmatrix}$

Step 2: Calculating $BC$

$BC = B \cdot C = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 2 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 0 & -2 \end{pmatrix}$

Step 3: Calculating $(A + B)C$

$A + B = \begin{pmatrix} 0 & 6 & 7 \\ -6 & 0 & 8 \end{pmatrix} + \begin{pmatrix} 1 & 0 & 1 \\ 1 & 2 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 6 & 8 \\ -5 & 2 & 8 \end{pmatrix}$

$(A + B)C = \begin{pmatrix} 1 & 6 & 8 \\ -5 & 2 & 8 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 0 & -2 \end{pmatrix}$

Step 4: Verify $(A + B)C = AC + BC$

Let's perform these calculations using Python.