The table summarizes the scoring of a football game between Team A and Team B. A touchdown (TD) is worth 6 points, a field goal (FG) is worth 3 points, a safety (S) is worth 2 points, and a point after touchdown (PAT) is worth 1 point. Use matrix multiplication to find the final score.%0D%0A%0D%0A %09TD %09FG%09S%09PAT%0D%0A Team A%09 3%09 2%09 1%09 0%0D%0A Team B%09 4%09 4%09 1%09 2%0D%0A%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0ATeam A: 35%0D%0ATeam B: 36%0D%0ATeam A : 35 Team B : 36%0D%0A%0D%0ATeam A: 36%0D%0ATeam B: 39%0D%0ATeam A : 36 Team B : 39%0D%0A%0D%0ATeam A: 26%0D%0ATeam B: 40%0D%0ATeam A : 26 Team B : 40%0D%0A%0D%0ATeam A: 17%0D%0ATeam B: 38
To find the final score using matrix multiplication, we can set up a matrix to represent the scoring values for each team. Let's label this matrix as M:
M = | 6 3 2 1 |
| 3 4 4 2 |
The first row of M represents the scoring values for Team A (TD, FG, S, PAT) and the second row represents the scoring values for Team B.
Next, we need to represent the number of times each scoring event occurred for each team as a column vector. Let's label these vectors as V:
V(Team A) = | 3 |
| 2 |
| 1 |
| 0 |
V(Team B) = | 4 |
| 4 |
| 1 |
| 2 |
Now, to find the final score, we can perform matrix multiplication between M and V for each team. The resulting vector will represent the total points scored by each team:
Final Score(Team A) = M x V(Team A)
Final Score(Team B) = M x V(Team B)
Let's calculate this:
Final Score(Team A) = | 6 3 2 1 | x | 3 | = | (6x3) + (3x2) + (2x1) + (1x0) | = | 35 |
| 3 4 4 2 |
Final Score(Team B) = | 6 3 2 1 | x | 4 | = | (6x4) + (3x4) + (2x1) + (1x2) | = | 36 |
| 3 4 4 2 |
Therefore, the final score is:
Team A: 35
Team B: 36
To find the final score using matrix multiplication, we can represent the scoring system as a matrix and multiply it by the scores of each team.
The scoring matrix is:
```
| 6 3 2 1 |
| 4 4 1 2 |
```
The score matrix for Team A is:
```
| 3 |
| 2 |
| 1 |
| 0 |
```
To find the final score for Team A, we multiply the scoring matrix by the score matrix for Team A:
```
| 6 3 2 1 | | 3 | | 35 |
| 4 4 1 2 | x | 2 | = | 44 |
```
So the final score for Team A is 35.
Similarly, the score matrix for Team B is:
```
| 4 |
| 4 |
| 1 |
| 2 |
```
To find the final score for Team B, we multiply the scoring matrix by the score matrix for Team B:
```
| 6 3 2 1 | | 4 | | 41 |
| 4 4 1 2 | x | 4 | = | 35 |
```
So the final score for Team B is 36.
Therefore, the final score for the game is:
Team A: 35
Team B: 36
To find the final score using matrix multiplication, we need to create a matrix containing the values for each scoring type (TD, FG, S, PAT) for each team:
Scoring matrix for Team A:
[6 3 2 1]
Scoring matrix for Team B:
[4 4 1 2]
We can then calculate the final score by multiplying these matrices:
Final Score = Scoring matrix for Team A x Scoring matrix for Team B
Final Score = [6 3 2 1] x [4 4 1 2]
Using matrix multiplication, the resulting matrix is:
Final Score = [6*4 + 3*4 + 2*1 + 1*2] = [24 + 12 + 2 + 2] = [40]
Therefore, the final score of the game is:
Team A: 40
Team B: 40