Question

Does the associative property for addition work for matrices? If A+(B+C)=(A+B)=C?
Explain Why Or Why not

Answer 1

Your turn. I did hte last two.

Answer 2

-_-

Answer 3

I dont know how to thts why im asking tk.

Answer 4

The Associative Property of Addition for Matrices states:

Let A, B, and C be m × n matrices. Then, (A + B) + C = A + (B + C).

Example:
\[A=\left[\begin{matrix}3 & 2 & 4 \\ -1 & 0 & -5\end{matrix}\right], B=\left[\begin{matrix}-2 & 3 & -1 \\ 4 & 2 & 0\end{matrix}\right], C=\left[\begin{matrix}8 & -1 & 5 \\ 6 & 1 & 2\end{matrix}\right]\]
Find (A + B) + C and A + (B + C)

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Find (A + B) + C:
\[\left( \left[\begin{matrix}3 & 2 & 4 \\ -1 & 0 & -5\end{matrix}\right]+\left[\begin{matrix}-2 & 3 & -1 \\ 4 & 2 & 0\end{matrix}\right] \right)+\left[\begin{matrix}8 & -1 & 5 \\ 6 & 1 & 2\end{matrix}\right]\]
\[=\left[\begin{matrix}1 & 5 & 3 \\ 3 & 2 & -5\end{matrix}\right]+\left[\begin{matrix}8 & -1 & 5 \\ 6 & 1 & 2\end{matrix}\right]\]
\[=\left[\begin{matrix}9 & 4 & 8 \\ 9 & 3 & -3\end{matrix}\right]\]

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Find A + (B + C):
\[\left[\begin{matrix}3 & 2 & 4 \\ -1 & 0 & -5\end{matrix}\right]+\left( \left[\begin{matrix}-2 & 3 & -1 \\ 4 & 2 & 0\end{matrix}\right]+ \left[\begin{matrix}8 & -1 & 5 \\ 6 & 1 & 2\end{matrix}\right]\right)\]
\[=\left[\begin{matrix}3 & 2 & 4 \\ -1 & 0 & -5\end{matrix}\right]+\left[\begin{matrix}6 & 2 & 4 \\ 10 & 3 & 2\end{matrix}\right]\]
\[=\left[\begin{matrix}9 & 4 & 8 \\ 9 & 3 & -3\end{matrix}\right]\]

If that makes sense. haha

Answer 5

ok so it does work but can u explain to me why?

Answer 6

Because basically (A+B)+C = A+(B+C) = A+B+C, the parentheses don't make a difference, in the end it's all adding A, B, and C.

Answer 7

Remember that Cumutative Property for Multiplication? "Examples can be constructed." This is very important. In fact, infinitely many exampels can be constructed, and yet it is not generally valid! A demonstration, or any number of examples, unless you can try ALL possibilities, will be insufficient.

There is NOTHING about matrix addition that does not have a direct analog in regular addition of Real Numbers when considered element-by-element. Done.