To determine whether the Distributive Property applies to scalar multiplication and addition of matrices, we need to evaluate and compare the following expressions:
1. [tex]\( A + dA \)[/tex]
2. [tex]\((c + d)A \)[/tex]
Given:
[tex]\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \][/tex]
[tex]\[ c = 3 \][/tex]
[tex]\[ d = 5 \][/tex]
### Step 1: Calculate [tex]\( dA \)[/tex]
First, we need to multiply matrix [tex]\( A \)[/tex] by the scalar [tex]\( d \)[/tex].
[tex]\[ dA = 5A = 5 \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 5 \cdot 1 & 5 \cdot 0 \\ 5 \cdot 0 & 5 \cdot 3 \end{bmatrix} = \begin{bmatrix} 5 & 0 \\ 0 & 15 \end{bmatrix} \][/tex]
### Step 2: Calculate [tex]\( A + dA \)[/tex]
Next, we add the original matrix [tex]\( A \)[/tex] to the matrix [tex]\( dA \)[/tex].
[tex]\[ A + dA = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} + \begin{bmatrix} 5 & 0 \\ 0 & 15 \end{bmatrix} = \begin{bmatrix} 1 + 5 & 0 + 0 \\ 0 + 0 & 3 + 15 \end{bmatrix} = \begin{bmatrix} 6 & 0 \\ 0 & 18 \end{bmatrix} \][/tex]
### Step 3: Calculate [tex]\( c + d \)[/tex]
We sum the scalars [tex]\( c \)[/tex] and [tex]\( d \)[/tex].
[tex]\[ c + d = 3 + 5 = 8 \][/tex]
### Step 4: Calculate [tex]\( (c + d)A \)[/tex]
Now, we need to multiply the original matrix [tex]\( A \)[/tex] by the scalar [tex]\( c + d \)[/tex].
[tex]\[ (c + d)A = 8A = 8 \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 8 \cdot 1 & 8 \cdot 0 \\ 8 \cdot 0 & 8 \cdot 3 \end{bmatrix} = \begin{bmatrix} 8 & 0 \\ 0 & 24 \end{bmatrix} \][/tex]
### Step 5: Compare [tex]\( A + dA \)[/tex] and [tex]\( (c + d)A \)[/tex]
Finally, we compare the results from steps 2 and 4:
[tex]\[ A + dA = \begin{bmatrix} 6 & 0 \\ 0 & 18 \end{bmatrix} \][/tex]
[tex]\[ (c + d)A = \begin{bmatrix} 8 & 0 \\ 0 & 24 \end{bmatrix} \][/tex]
We see that:
[tex]\[ \begin{bmatrix} 6 & 0 \\ 0 & 18 \end{bmatrix} \neq \begin{bmatrix} 8 & 0 \\ 0 & 24 \end{bmatrix} \][/tex]
### Conclusion
The matrices [tex]\( A + dA \)[/tex] and [tex]\( (c + d)A \)[/tex] are not equal. Therefore, the Distributive Property does not hold in this case for scalar multiplication and addition of matrices.
