To determine which property is illustrated in the given matrix addition, consider the specific matrices and their resultant matrix:
[tex]\[
\left[\begin{array}{ccc}
6 & -8 & 1 \\
0 & 2 & -19
\end{array}\right] + \left[\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] = \left[\begin{array}{ccc}
6 & -8 & 1 \\
0 & 2 & -19
\end{array}\right]
\][/tex]
We need to compare each property to see which one is applicable here:
1. Inverse Property: This property states that for any matrix [tex]\( A \)[/tex], there exists a matrix [tex]\( B \)[/tex] such that [tex]\( A + B = 0 \)[/tex] where [tex]\( B = -A \)[/tex]. In this case, [tex]\( B \)[/tex] would be the additive inverse of [tex]\( A \)[/tex]. However, adding the given matrices does not result in a zero matrix. Hence, the inverse property does not apply.
2. Identity Property: This property states that for any matrix [tex]\( A \)[/tex], adding the identity matrix for addition (a matrix with all elements as 0) to [tex]\( A \)[/tex] results in the matrix [tex]\( A \)[/tex] itself. In this problem, we observe that adding a [tex]\([2 \times 3]\)[/tex] matrix filled with zeros to the original matrix results in the original matrix. This verifies the identity property.
3. Commutative Property: This property states that the order of addition does not affect the result. For matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex], [tex]\( A + B = B + A \)[/tex]. While this property is generally true for matrix addition, it doesn't specifically explain why the matrix remains unchanged in this scenario.
4. Associative Property: This property states that the grouping of matrices does not affect the sum. For matrices [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex], [tex]\((A + B) + C = A + (B + C)\)[/tex]. This property pertains to how we group the addition of multiple matrices but does not explain the invariance observed in this particular addition.
Given the analysis, the property illustrated by the matrix equation
[tex]\[
\left[\begin{array}{ccc}
6 & -8 & 1 \\
0 & 2 & -19
\end{array}\right] + \left[\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0
\end{array}\right] = \left[\begin{array}{ccc}
6 & -8 & 1 \\
0 & 2 & -19
\end{array}\right]
\][/tex]
is the identity property. Therefore, the correct answer is:
Identity Property
