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a).           Varuni, Kamani and Thilini are three women engaged in businesses such as fruit juice outlets, bookstores and textile shops. Varuni has 05 fruit juice outlets,03 bookstores and 02 textile shops. Kamani has 10 fruit juice outlets, 08 bookstores and 05 textile shops. Thilini has 10 fruit juice outlets, 08 bookstores and 05 textile shops.
 
 
                      i).          Express the above information in matrix form.

                    ii).          Explain how the matrix you found differs from a triangular matrix.

(04 Marks

To tackle this problem, we need to first represent the information given in matrix form and then analyze how this matrix differs from a triangular matrix.

### Part (i): Express the information in matrix form

Let's define the matrix \( A \) where each row represents the number of businesses for Varuni, Kamani, and Thilini respectively, and the columns represent fruit juice outlets, bookstores, and textile shops.

Given the dataTo tackle this problem, we need to first represent the information given in matrix form and then analyze how this matrix differs from a triangular matrix.

### Part (i): Express the information in matrix form

Let's define the matrix \( A \) where each row represents the number of businesses for Varuni, Kamani, and Thilini respectively, and the columns represent fruit juice outlets, bookstores, and textile shops.

Given the 
- Varuni has 5 fruit juice outlets, 3 bookstores, and 2 textile shops.
- Kamani has 10 fruit juice outlets, 8 bookstores, and 5 textile shops.
- Thilini has the same numbers as Kamani: 10 fruit juice outlets, 8 bookstores, and 5 textile shops.

The matrix \( A \) is:

\[
A = \begin{pmatrix}
5 & 3 & 2 \\
10 & 8 & 5 \\
10 & 8 & 5
\end{pmatrix}
\]

### Part (ii): How does this matrix differ from a triangular matrix?

A triangular matrix is a special type of square matrix where all the elements above (or below) the main diagonal are zeros. There are two types of triangular matrices:
1. **Upper triangular matrix**: All elements below the main diagonal are zero.
2. **Lower triangular matrix**: All elements above the main diagonal are zero.

In contrast:
- The matrix \( A \) is not a triangular matrix because it contains non-zero elements both above and below the main diagonal.
- The main diagonal of matrix \( A \) consists of elements \( 5, 8, 5 \), and there are non-zero elements both in the upper and lower parts of the matrix relative to this diagonal.

Thus, the matrix \( A \) differs from a triangular matrix because it has non-zero elements in positions that would be zeros in a triangular matrix.

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Would you like to explore more details on triangular matrices or any other related topics? Here are some further questions to expand your understanding:

1. What are the properties of an identity matrix?
2. How do we determine if a matrix is invertible?
3. Can a triangular matrix be diagonalized? If yes, how?
4. What are the applications of triangular matrices in solving linear equations?
5. How would you calculate the determinant of a triangular matrix?

**Tip:** In matrix operations, understanding special types of matrices (like triangular matrices) can simplify many calculations, especially in solving systems of equations.

Learn how to represent business data using matrices and understand the concept of triangular matrices. This problem illustrates the use of matrices to organize and analyze business information. Suitable for students in Grades 7-9.

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