Answer :
To determine which property is demonstrated in the given matrix addition:
[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 will analyze the nature of matrix addition and relevant properties:
### Step-by-Step Analysis:
1. Matrix on the Left Side:
[tex]\[ \left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right] \][/tex]
2. Matrix Being Added (Right Side Matrix):
[tex]\[ \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \][/tex]
3. Observation of the Sum:
Adding these two matrices, each corresponding element-wise addition will be:
[tex]\[ \left[\begin{array}{ccc} 6 + 0 & -8 + 0 & 1 + 0 \\ 0 + 0 & 2 + 0 & -19 + 0 \end{array}\right] = \left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right] \][/tex]
4. Conclusion of the Property:
The resulting matrix remains unchanged as [tex]\(\left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right]\)[/tex].
The property states that any matrix added to a zero matrix (a matrix with all elements 0) remains the same. This is known as the Identity Property of Matrix Addition. This property signifies that the zero matrix acts as the additive identity for matrix addition.
### Final Answer:
The property shown in the given matrix addition is the identity property.
[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 will analyze the nature of matrix addition and relevant properties:
### Step-by-Step Analysis:
1. Matrix on the Left Side:
[tex]\[ \left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right] \][/tex]
2. Matrix Being Added (Right Side Matrix):
[tex]\[ \left[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \][/tex]
3. Observation of the Sum:
Adding these two matrices, each corresponding element-wise addition will be:
[tex]\[ \left[\begin{array}{ccc} 6 + 0 & -8 + 0 & 1 + 0 \\ 0 + 0 & 2 + 0 & -19 + 0 \end{array}\right] = \left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right] \][/tex]
4. Conclusion of the Property:
The resulting matrix remains unchanged as [tex]\(\left[\begin{array}{ccc} 6 & -8 & 1 \\ 0 & 2 & -19 \end{array}\right]\)[/tex].
The property states that any matrix added to a zero matrix (a matrix with all elements 0) remains the same. This is known as the Identity Property of Matrix Addition. This property signifies that the zero matrix acts as the additive identity for matrix addition.
### Final Answer:
The property shown in the given matrix addition is the identity property.
