Question

Given the matrices 

`A=[[2,1,1],[3,-1,0],[0,2,4]]` , `B=[[9,7,-1],[3,5,4],[2,1,6]]`  `and  C=[[2,-4,3],[1,-1,0],[9,4,5]]`

Verify that (A + B) + C = A + (B + C).

 

Answer 1

LaTeX

\[Here, \]
\[ LHS = \left( A + B \right) + C\]
\[ = \left( \begin{bmatrix}2 & 1 & 1 \\ 3 & - 1 & 0 \\ 0 & 2 & 4\end{bmatrix} + \begin{bmatrix}9 & 7 & - 1 \\ 3 & 5 & 4 \\ 2 & 1 & 6\end{bmatrix} \right) + \begin{bmatrix}2 & - 4 & 3 \\ 1 & - 1 & 0 \\ 9 & 4 & 5\end{bmatrix}\]
\[ = \left( \begin{bmatrix}2 + 9 & 1 + 7 & 1 - 1 \\ 3 + 3 & - 1 + 5 & 0 + 4 \\ 0 + 2 & 2 + 1 & 4 + 6\end{bmatrix} \right) + \begin{bmatrix}2 & - 4 & 3 \\ 1 & - 1 & 0 \\ 9 & 4 & 5\end{bmatrix}\]
\[ = \begin{bmatrix}11 & 8 & 0 \\ 6 & 4 & 4 \\ 2 & 3 & 10\end{bmatrix} + \begin{bmatrix}2 & - 4 & 3 \\ 1 & - 1 & 0 \\ 9 & 4 & 5\end{bmatrix}\]
\[ = \begin{bmatrix}11 + 2 & 8 - 4 & 0 + 3 \\ 6 + 1 & 4 - 1 & 4 + 0 \\ 2 + 9 & 3 + 4 & 10 + 5\end{bmatrix}\]
\[ = \begin{bmatrix}13 & 4 & 3 \\ 7 & 3 & 4 \\ 11 & 7 & 15\end{bmatrix}\]
\[RHS = A + \left( B + C \right)\]
\[ = \begin{bmatrix}2 & 1 & 1 \\ 3 & - 1 & 0 \\ 0 & 2 & 4\end{bmatrix} + \left( \begin{bmatrix}9 & 7 & - 1 \\ 3 & 5 & 4 \\ 2 & 1 & 6\end{bmatrix} + \begin{bmatrix}2 & - 4 & 3 \\ 1 & - 1 & 0 \\ 9 & 4 & 5\end{bmatrix} \right)\]
\[ = \begin{bmatrix}2 & 1 & 1 \\ 3 & - 1 & 0 \\ 0 & 2 & 4\end{bmatrix} + \left( \begin{bmatrix}9 + 2 & 7 - 4 & - 1 + 3 \\ 3 + 1 & 5 - 1 & 4 + 0 \\ 2 + 9 & 1 + 4 & 6 + 5\end{bmatrix} \right)\]
\[ = \begin{bmatrix}2 & 1 & 1 \\ 3 & - 1 & 0 \\ 0 & 2 & 4\end{bmatrix} + \begin{bmatrix}11 & 3 & 2 \\ 4 & 4 & 4 \\ 11 & 5 & 11\end{bmatrix}\]
\[ = \begin{bmatrix}2 + 11 & 1 + 3 & 1 + 2 \\ 3 + 4 & - 1 + 4 & 0 + 4 \\ 0 + 11 & 2 + 5 & 4 + 11\end{bmatrix}\]
\[ = \begin{bmatrix}13 & 4 & 3 \\ 7 & 3 & 4 \\ 11 & 7 & 15\end{bmatrix}\]
\[ \therefore LHS = RHS\]