The given matrix addition expression demonstrates a specific property of matrix addition. To determine which property is being shown, let's review the matrix addition and the properties provided as options.
Starting with the left-hand side of the equation:
[tex]\[
\left[\begin{array}{c}
5 \\
-7 \\
6.2 \\
12
\end{array}\right]+\left(\left[\begin{array}{c}
-1 \\
0.4 \\
-9.9 \\
0
\end{array}\right]+\left[\begin{array}{c}
0 \\
0 \\
-8.5 \\
2
\end{array}\right]\right)
\][/tex]
And comparing it to the right-hand side:
[tex]\[
\left(\left[\begin{array}{c}
5 \\
-7 \\
6.2 \\
12
\end{array}\right]+\left[\begin{array}{c}
-1 \\
0.4 \\
-9.9 \\
0
\end{array}\right]\right)+\left[\begin{array}{c}
0 \\
0 \\
-8.5 \\
2
\end{array}\right]
\][/tex]
What we see here is that the order of the matrices themselves has not changed, but the grouping of the matrices in the addition operation has changed. The matrices:
[tex]\[
\left[\begin{array}{c}
5 \\
-7 \\
6.2 \\
12
\end{array}\right],
\left[\begin{array}{c}
-1 \\
0.4 \\
-9.9 \\
0
\end{array}\right],
\left[\begin{array}{c}
0 \\
0 \\
-8.5 \\
2
\end{array}\right]
\][/tex]
are grouped in different ways, yet this re-grouping does not affect the outcome of the matrix addition.
This property is known as the associative property, which states that how the elements are grouped in an addition operation does not change the result. Therefore, the correct answer is:
associative property
