Question

Which equation illustrates the matrix associative property of addition

Answer 1

c

cuz it is

first of all 4,3 + 0,0 = 4,3

2,1+0,0 = 2,1

and which is the answer

Answer 2

The correct option is d.

After performing the calculations, we find that both sides of Equation D yield the same result:

1. Left-hand side:
`\( \left[\begin{array}{l}4 \\ 2\end{array}\right] + \left( \left[\begin{array}{l}3 \\ 5\end{array}\right] + \left[\begin{array}{l}1 \\ 3\end{array}\right] \right) = \left[\begin{array}{l}8 \\ 10\end{array}\right] \)`

2. Right-hand side:
`\( \left( \left[\begin{array}{l}4 \\ 2\end{array}\right] + \left[\begin{array}{l}3 \\ 5\end{array}\right] \right) + \left[\begin{array}{l}1 \\ 3\end{array}\right] = \left[\begin{array}{l}8 \\ 10\end{array}\right] \)`

The associative property of matrix addition states that the way matrices are grouped during addition does not change their sum. This means for any three matrices A, B, and C, the equation A + (B + C) = (A + B) + C holds.

Let's analyze the provided options:

A.
`\( 4\left(\left[\begin{array}{l}3 \\ 5\end{array}\right]+\left[\begin{array}{l}1 \\ 3\end{array}\right]\right)=4\left[\begin{array}{l}3 \\ 5\end{array}\right]+4\left[\begin{array}{l}1 \\ 3\end{array}\right] \)` - This equation illustrates the distributive property, not the associative property.

B.
`\( \left[\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right]+\left[\begin{array}{ll}8 & 6 \\ 2 & 4\end{array}\right]=\left[\begin{array}{ll}8 & 6 \\ 2 & 4\end{array}\right]+\left[\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right] \)` - This shows the commutative property of addition, not the associative property.

C.
`\( \left[\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right]+\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=\left[\begin{array}{ll}4 & 3 \\ 2 & 1\end{array}\right] \)` - This demonstrates the addition of a zero matrix (identity element for addition), not the associative property.

D.
`\( \left[\begin{array}{l}4 \\ 2\end{array}\right]+\left(\left[\begin{array}{l}3 \\ 5\end{array}\right]+\left[\begin{array}{l}1 \\ 3\end{array}\right]\right)=\left(\left[\begin{array}{l}4 \\ 2\end{array}\right]+\left[\begin{array}{l}3 \\ 5\end{array}\right]\right)+\left[\begin{array}{l}1 \\ 3\end{array}\right] \)` - This equation is an example of the associative property of matrix addition.

So, the answer is D.

Let's verify this by performing the calculations step by step:

1. Calculate
`\( \left[\begin{array}{l}3 \\ 5\end{array}\right] + \left[\begin{array}{l}1 \\ 3\end{array}\right] \)`

2. Add this result to
`\( \left[\begin{array}{l}4 \\ 2\end{array}\right] \)`

3. Perform the same operations on the right-hand side of the equation

4. Compare both sides to confirm the associative property

Let's calculate these steps:

Sure, let's perform these calculations step by step to verify the associative property of matrix addition as illustrated in Option D:

1. Calculate
`\( \left[\begin{array}{l}3 \\ 5\end{array}\right] + \left[\begin{array}{l}1 \\ 3\end{array}\right] \)`:

`\[ \left[\begin{array}{l}3 \\ 5\end{array}\right] + \left[\begin{array}{l}1 \\ 3\end{array}\right] = \left[\begin{array}{l}3 + 1 \\ 5 + 3\end{array}\right] = \left[\begin{array}{l}4 \\ 8\end{array}\right] \]`

2. Add this result to
`\( \left[\begin{array}{l}4 \\ 2\end{array}\right] \)` for the left-hand side of the equation:

`\[ \left[\begin{array}{l}4 \\ 2\end{array}\right] + \left[\begin{array}{l}4 \\ 8\end{array}\right] = \left[\begin{array}{l}4 + 4 \\ 2 + 8\end{array}\right] = \left[\begin{array}{l}8 \\ 10\end{array}\right] \]`

3. Perform the same operations on the right-hand side of the equation:

a. First, add
`\( \left[\begin{array}{l}4 \\ 2\end{array}\right] \) and \( \left[\begin{array}{l}3 \\ 5\end{array}\right] \)`:

`\[ \left[\begin{array}{l}4 \\ 2\end{array}\right] + \left[\begin{array}{l}3 \\ 5\end{array}\right] = \left[\begin{array}{l}4 + 3 \\ 2 + 5\end{array}\right] = \left[\begin{array}{l}7 \\ 7\end{array}\right] \]`

b. Now, add the result to \( \left[\begin{array}{l}1 \\ 3\end{array}\right] \):

`\[ \left[\begin{array}{l}7 \\ 7\end{array}\right] + \left[\begin{array}{l}1 \\ 3\end{array}\right] = \left[\begin{array}{l}7 + 1 \\ 7 + 3\end{array}\right] = \left[\begin{array}{l}8 \\ 10\end{array}\right] \]`

4. Compare both sides to confirm the associative property:

Both the left-hand side and the right-hand side of the equation yield the same result:

`\[ \left[\begin{array}{l}8 \\ 10\end{array}\right] = \left[\begin{array}{l}8 \\ 10\end{array}\right] \]`

Since both sides are equal, this confirms that Equation D accurately represents the associative property of matrix addition.