To solve for the entries [tex]\( c_{41} \)[/tex] in matrices [tex]\( C \)[/tex] and [tex]\( D \)[/tex], we need to understand the operations involved in the given expressions.
1. First, we will evaluate Matrix [tex]\( C \)[/tex]:
[tex]\[
C = (A - B) + A
\][/tex]
This expression can be broken down into two parts:
- Calculate [tex]\( A - B \)[/tex]
- Sum the result with [tex]\( A \)[/tex]
2. To find the value for [tex]\( c_{41} \)[/tex] in [tex]\( C \)[/tex], which is the entry in the 4th row and 1st column of [tex]\( C \)[/tex], we perform the following steps based on matrix indices:
3. We will then find [tex]\( c_{41} \)[/tex] in the expression:
[tex]\[
D = (A + B) - A
\][/tex]
This expression simplifies directly:
[tex]\[
D = B
\][/tex]
Let’s input the correct values in the boxes given the final results:
[tex]\[
A=\left[\begin{array}{ccc}
-5 & 3 & 8 \\
3 & 6 & -5 \\
5 & -9 & 0 \\
7 & 3 & 4
\end{array}\right] \quad B=\left[\begin{array}{ccc}
-7 & -8 & -5 \\
7 & 9 & 2 \\
2 & 5 & -7 \\
2 & 8 & -7
\end{array}\right]
\][/tex]
If matrix [tex]\( C \)[/tex] represents [tex]\( (A - B) + A \)[/tex]:
[tex]\[
C = \left[\begin{array}{ccc}
2 & 11 & 13 \\
-4 & -3 & -7 \\
3 & -14 & 7 \\
12 & -4 & 11
\end{array}\right]
\][/tex]
If matrix [tex]\( D \)[/tex] represents [tex]\( (A + B) - A \)[/tex]:
[tex]\[
D = B
= \left[\begin{array}{ccc}
-7 & -8 & -5 \\
7 & 9 & 2 \\
2 & 5 & -7 \\
2 & 8 & -7
\end{array}\right]
\][/tex]
Thus, the entry [tex]\( c_{41} \)[/tex] in [tex]\( C \)[/tex] is:
[tex]\[
12
\][/tex]
And the corresponding entry in [tex]\( D \)[/tex] is:
[tex]\[
2
\][/tex]
So, the answers are:
If matrix [tex]\( C \)[/tex] represents [tex]\( (A-B) + A \)[/tex], the value of the entry represented by [tex]\( c_{41} \)[/tex] is [tex]\( \boxed{12} \)[/tex] and the corresponding entry in [tex]\( (A+B)-A \)[/tex] is [tex]\( \boxed{2} \)[/tex].
