✅ Matrices and Sparse Matrices

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✅ 1. Matrix (Matrices)

✅ Definition

A Matrix is a rectangular arrangement of elements (numbers/data) in the form of rows and columns.

✅ It is written as:

[
A = [a_{ij}]
]

Where:

• i = row number
• j = column number

📌 Example of a Matrix (3×3):
[
A =
\begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9
\end{bmatrix}
]

✅ In programming, matrices are represented using 2D Arrays.

Example:

int A[3][3];

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✅ Types of Matrices (Important)

✅ 1. Row Matrix

Matrix having only one row.

Example (1×3):
[
[ 10\ \ 20\ \ 30 ]
]

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✅ 2. Column Matrix

Matrix having only one column.

Example (3×1):
[
\begin{bmatrix}
10\
20\
30
\end{bmatrix}
]

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✅ 3. Square Matrix

Matrix having same number of rows and columns.

Example (3×3)

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✅ 4. Diagonal Matrix

All elements except diagonal are 0.

Example:
[
\begin{bmatrix}
5 & 0 & 0\
0 & 3 & 0\
0 & 0 & 1
\end{bmatrix}
]

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✅ 5. Identity Matrix

Diagonal elements are 1, others are 0.

Example:
[
\begin{bmatrix}
1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 1
\end{bmatrix}
]

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✅ 6. Zero (Null) Matrix

All elements are 0.

Example:
[
\begin{bmatrix}
0 & 0\
0 & 0
\end{bmatrix}
]

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✅ Applications of Matrices

Matrices are used in:
✅ Computer graphics
✅ Image processing
✅ Scientific calculations
✅ Engineering problems
✅ Data representation in tables
✅ Graphs (Adjacency matrix)

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✅ 2. Sparse Matrix

✅ Definition

A Sparse Matrix is a matrix in which most of the elements are zero.

✅ Condition:
If number of zero elements is greater than non-zero elements, then it is called a Sparse Matrix.

📌 Example:
[
\begin{bmatrix}
0 & 0 & 5 & 0\
0 & 0 & 0 & 0\
0 & 8 & 0 & 0\
0 & 0 & 0 & 0
\end{bmatrix}
]

Here:

• Non-zero elements = 2 (5 and 8)
• Remaining are zeros → many zeros → ✅ Sparse Matrix

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✅ Why Sparse Matrix is Used?

Storing sparse matrix in normal 2D array wastes memory because it stores too many zeros.

✅ So we use special representation to save:

• Memory (Space)
• Processing time

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✅ 3. Sparse Matrix Representation

To store only non-zero elements, we use 3-tuple (Triplet) representation.

✅ Triplet Form / 3-Tuple Representation

In this method, sparse matrix is stored using a table with 3 columns:

✅ Columns are:

1. Row number
2. Column number
3. Value

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✅ Example of Sparse Matrix

Matrix (4×4):
[
\begin{bmatrix}
0 & 0 & 3 & 0\
0 & 0 & 0 & 0\
0 & 5 & 0 & 0\
0 & 0 & 0 & 7
\end{bmatrix}
]

Non-zero values are:

• 3 at (0,2)
• 5 at (2,1)
• 7 at (3,3)

✅ Triplet Table:

Row	Col	Value
4	4	3
0	2	3
2	1	5
3	3	7

📌 First row stores:

• Total rows
• Total columns
• Total non-zero elements

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✅ 4. Advantages of Sparse Matrix

✅ Benefits:

• Saves memory space (stores only non-zero values)
• Fast processing for big matrices
• Efficient storage representation

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✅ 5. Disadvantages of Sparse Matrix

❌ Limitations:

• More complex than normal matrix
• Accessing elements becomes slower because searching is needed
• Not suitable if non-zero values are high

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✅ 6. Difference Between Matrix and Sparse Matrix

Feature	Matrix	Sparse Matrix
Definition	Rows and columns data	Mostly zero values
Storage	Normal 2D array	Special compact form
Memory Usage	High for large matrices	Very less
Best For	General problems	Large matrix with many zeros
Example	Marks table	Adjacency matrix of sparse graph

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✅ Conclusion

A matrix is a 2D structure of elements arranged in rows and columns and is commonly stored using 2D arrays. A sparse matrix is a special matrix in which most elements are zero, and it is stored efficiently using triplet (row, column, value) representation to save memory and improve performance.

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