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Question
Let `A=[[1,1,1],[3,3,3]],B=[[3,1],[5,2],[-2,4]]` and `C=[[4,2],[-3,5],[5,0]]`Verify that AB = AC though B ≠ C, A ≠ O.
Solution
\[Here, \]
\[A = \begin{bmatrix}1 & 1 & 1 \\ 3 & 3 & 3\end{bmatrix}, B = \begin{bmatrix}3 & 1 \\ 5 & 2 \\ - 2 & 4\end{bmatrix} \text{and C} = \begin{bmatrix}4 & 2 \\ - 3 & 5 \\ 5 & 0\end{bmatrix}\]
\[Now, \]
\[AB = \begin{bmatrix}1 & 1 & 1 \\ 3 & 3 & 3\end{bmatrix}\begin{bmatrix}3 & 1 \\ 5 & 2 \\ - 2 & 4\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}3 + 5 - 2 & 1 + 2 + 4 \\ 9 + 15 - 6 & 3 + 6 + 12\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}6 & 7 \\ 18 & 21\end{bmatrix}\]
\[\]
\[AC = \begin{bmatrix}1 & 1 & 1 \\ 3 & 3 & 3\end{bmatrix}\begin{bmatrix}4 & 2 \\ - 3 & 5 \\ 5 & 0\end{bmatrix}\]
\[ \Rightarrow AC = \begin{bmatrix}4 - 3 + 5 & 2 + 5 + 0 \\ 12 - 9 + 15 & 6 + 15 + 0\end{bmatrix}\]
\[ \Rightarrow AC = \begin{bmatrix}6 & 7 \\ 18 & 21\end{bmatrix}\]
So, AB = AC though B ≠ C , A ≠ O.
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