91Ó°ÊÓ

First, we need to determine the matrix \(B + C\). Using matrix addition, we add corresponding elements from matrices \(B\) and \(C\):\[B + C = \left[\begin{array}{rrr}6+1 & 2+4 & 7-3 \3+8 & -4+1 & -5-1 \7+4 & 1+6 & 0-2\end{array}\right] =\left[\begin{array}{rrr}7 & 6 & 4 \11 & -3 & -6 \11 & 7 & -2\end{array}\right]\]

Now that we have \(B+C\), we can find \(A(B+C)\) by multiplying matrices \(A\) and \(B+C\): \[A(B+C) = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{rrr}7 & 6 & 4 \11 & -3 & -6 \11 & 7 & -2\end{array}\right] \]Calculating each element, we have:- First row: \((2 \times 7) + (-1 \times 11) + (3 \times 11), (2 \times 6) + (-1 \times -3) + (3 \times 7), (2 \times 4) + (-1 \times -6) + (3 \times -2)\)- Second row: \((1 \times 7) + (3 \times 11) + (-5 \times 11), (1 \times 6) + (3 \times -3) + (-5 \times 7), (1 \times 4) + (3 \times -6) + (-5 \times -2)\)- Third row: \((0 \times 7) + (-2 \times 11) + (1 \times 11), (0 \times 6) + (-2 \times -3) + (1 \times 7), (0 \times 4) + (-2 \times -6) + (1 \times -2)\)Calculate these values to obtain the resulting matrix.

Next, calculate \(AB\) by multiplying matrices \(A\) and \(B\):\[AB = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{rrr}6 & 2 & 7 \3 & -4 & -5 \7 & 1 & 0\end{array}\right] \]Calculate each element:- First row: \((2 \times 6) + (-1 \times 3) + (3 \times 7), (2 \times 2) + (-1 \times -4) + (3 \times 1), (2 \times 7) + (-1 \times -5) + (3 \times 0)\)- Second row: \((1 \times 6) + (3 \times 3) + (-5 \times 7), (1 \times 2) + (3 \times -4) + (-5 \times 1), (1 \times 7) + (3 \times -5) + (-5 \times 0)\)- Third row: \((0 \times 6) + (-2 \times 3) + (1 \times 7), (0 \times 2) + (-2 \times -4) + (1 \times 1), (0 \times 7) + (-2 \times -5) + (1 \times 0)\) Perform these calculations to get the resulting matrix.

Now, find the product \(AC\):\[AC = \left[\begin{array}{rrr}2 & -1 & 3 \1 & 3 & -5 \0 & -2 & 1 \end{array}\right] \times \left[\begin{array}{lll}1 & 4 & -3 \8 & 1 & -1 \4 & 6 & -2\end{array}\right] \]Calculate each element:- First row: \((2 \times 1) + (-1 \times 8) + (3 \times 4), (2 \times 4) + (-1 \times 1) + (3 \times 6), (2 \times -3) + (-1 \times -1) + (3 \times -2)\)- Second row: \((1 \times 1) + (3 \times 8) + (-5 \times 4), (1 \times 4) + (3 \times 1) + (-5 \times 6), (1 \times -3) + (3 \times -1) + (-5 \times -2)\)- Third row: \((0 \times 1) + (-2 \times 8) + (1 \times 4), (0 \times 4) + (-2 \times 1) + (1 \times 6), (0 \times -3) + (-2 \times -1) + (1 \times -2)\)After computing, you will obtain the resulting matrix.

With the results from steps 3 and 4, compute the sum \(AB + AC\). Add corresponding elements from the matrices \(AB\) and \(AC\):\[AB + AC = \left[ Result \ AB \right] + \left[ Result \ AC \right]\]Perform the matrix addition by adding each corresponding element.

Now compare the matrices obtained from \(A(B+C)\) and \(AB + AC\). According to the distributive property of matrix multiplication, these two should be the same if calculations are correct.