There are 4 routes from city A to city B, 5 routes from city B to city C and 3 routes from city C to city A. Convert the above information into matrix form.

Answer

To convert the given information about the routes between cities A, B, and C into matrix form, we follow these steps:

### Step 1: Identify the cities and their routes
We have three cities:
- City A
- City B
- City C

The routes between the cities are:
- From City A to City B: 4 routes
- From City B to City C: 5 routes
- From City C to City A: 3 routes

### Step 2: Set up the matrix
Since we have three cities, we will create a 3x3 matrix. The rows will represent the starting city, and the columns will represent the destination city.

### Step 3: Fill in the matrix
We will fill in the matrix based on the number of routes:
- The entry in the first row and second column (A to B) will be 4.
- The entry in the second row and third column (B to C) will be 5.
- The entry in the third row and first column (C to A) will be 3.
- All other entries will be 0, as there are no direct routes between the other pairs of cities.

### Step 4: Write the matrix
The matrix can be represented as follows:

\[ \begin{bmatrix} 0 & 4 & 0 \\ 0 & 0 & 5 \\ 3 & 0 & 0 \end{bmatrix} \]

### Final Matrix
Thus, the final matrix representing the routes is:

\[ \begin{bmatrix} 0 & 4 & 0 \\ 0 & 0 & 5 \\ 3 & 0 & 0 \end{bmatrix} \]