You are on the right track, but you made a small mistake in interpreting the given data. Let's correct it and find the matrices.
Let x be the number of days (4 days) and y be the number of kilometers driven (560 km).
The charges of each company can be represented as follows:
Company A: 660x
Company B: 48y
Company C: 300x + 150y
The 3x1 matrix for the charges would look like this:
| 660x |
C = | 48y |
| 300x+150y |
Now, we need to find the matrix that when multiplied by C gives the charges of the three companies. That matrix should have two elements: x (number of days) and y (number of kilometers driven). So, the 2x1 matrix would look like this:
| 4 |
D = |560 |
Now let's find the product of these two matrices to find the charges from each company:
E = CD
| 660*4 | | 4 | | 2640 |
E = | 48*560 | * |560 | = | 26880 |
| 300*4+150*560 | | 97800 |
The matrix E now contains the charges for each company:
Company A: $2640
Company B: $26880
Company C: $97800
To save cost, John should hire the car from Company A.