A is a square matrix of order n.
l = maximum number of distinct entries if A is a triangular matrix
m = maximum number of distinct entries if A is a diagonal matrix
p = minimum number of zeroes if A is a triangular matrix
If `l + 5 = p + 2 m`, find the order of the matrix.

Explanation

To solve the problem, we need to find the order of the square matrix \( A \) given the relationships between the maximum number of distinct entries in triangular and diagonal matrices, as well as the minimum number of zeroes in a triangular matrix.

### Step 1: Define the variables
1. **l** = maximum number of distinct entries if \( A \) is a triangular matrix.
2. **m** = maximum number of distinct entries if \( A \) is a diagonal matrix.
3. **p** = minimum number of zeroes if \( A \) is a triangular matrix.

### Step 2: Calculate \( l \)
For a triangular matrix of order \( n \):
- The entries above the diagonal can be any distinct non-zero values.
- The entries on the diagonal can also be distinct non-zero values.
- The total number of entries in a triangular matrix is given by the sum of the first \( n \) natural numbers plus one for the zero.

Thus, we have:
\[ l = \frac{n(n + 1)}{2} + 1 \]

### Step 3: Calculate \( m \)
For a diagonal matrix of order \( n \):
- There are \( n \) diagonal entries, and all other entries are zero.
- The maximum number of distinct entries is the \( n \) diagonal entries plus one for the zero.

Thus, we have:
\[ m = n + 1 \]

### Step 4: Calculate \( p \)
For a triangular matrix of order \( n \):
- The minimum number of zeroes occurs when we fill the lower triangular part with non-zero values.
- The number of zeroes is equal to the number of entries below the diagonal, which is given by \( \frac{n(n - 1)}{2} \).

Thus, we have:
\[ p = \frac{n(n - 1)}{2} \]

### Step 5: Set up the equation
According to the problem, we have the equation:
\[ l + 5 = p + 2m \]

Substituting the expressions for \( l \), \( m \), and \( p \):
\[ \left(\frac{n(n + 1)}{2} + 1\right) + 5 = \left(\frac{n(n - 1)}{2}\right) + 2(n + 1) \]

### Step 6: Simplify the equation
Simplifying the left-hand side:
\[ \frac{n(n + 1)}{2} + 6 \]

Simplifying the right-hand side:
\[ \frac{n(n - 1)}{2} + 2n + 2 \]

Now, equate both sides:
\[ \frac{n(n + 1)}{2} + 6 = \frac{n(n - 1)}{2} + 2n + 2 \]

### Step 7: Eliminate the fractions
Multiply the entire equation by 2 to eliminate the fractions:
\[ n(n + 1) + 12 = n(n - 1) + 4n + 4 \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ n^2 + n + 12 = n^2 - n + 4n + 4 \]
\[ n + 12 = 3n + 4 \]

### Step 9: Solve for \( n \)
Rearranging further:
\[ 12 - 4 = 3n - n \]
\[ 8 = 2n \]
\[ n = 4 \]

### Conclusion
The order of the matrix \( A \) is \( n = 4 \).