Question

The number of additions and multiplications involved in performing Gaussian elimination on any \( n \times n \) upper triangular matrix is of the order:

• A. \( O(n) \)
• B. \( O(n^2) \)
• C. \( O(n^3) \)
• D. \( O(n^4) \)

Hint:

When analyzing the complexity of algorithms like Gaussian elimination, consider the number of operations performed per row and the total number of rows processed. The complexity is often quadratic for matrix operations.

Answer 1

The Correct Option is B

Gaussian elimination is a method for solving systems of linear equations. In the context of an \( n \times n \) upper triangular matrix, we perform row operations to eliminate variables. The number of operations required to perform Gaussian elimination depends on the size of the matrix and the number of elements we need to process.
For an upper triangular matrix, the process involves eliminating variables by subtracting multiples of one row from another. The number of operations per row is linear in the size of the matrix. Specifically:
In the first row, we perform \( n-1 \) operations (multiplications and additions).
In the second row, we perform \( n-2 \) operations.
Continuing this pattern, in the last row, we perform 1 operation.
The total number of operations is the sum of the operations for each row:
\[ (n-1) + (n-2) + \cdots + 1 = \frac{n(n-1)}{2} = O(n^2). \]
Thus, the number of additions and multiplications involved in Gaussian elimination is of the order \( O(n^2) \).