Question

Assuming that one person can add two numbers in \(t_{e}\) time, how long will it take that person to add two \(n \times n\) matrices considering the operation of addition as the basic operation? Justify your answer.

Short answer

Adding two \(n \times n\) matrices will require \(n^2\) additions, so it will take \(n^2 \times t_{e}\) time, assuming that each addition operation takes \(t_{e}\) time. The time complexity of this operation is \(O(n^2)\).

Step-by-step solution

Understanding matrix addition

In order to add two matrices, each element in one matrix must be added to the corresponding element in the other matrix. When the matrices are square and of size \(n \times n\), they each have \(n^2\) elements.

Counting the basic operations

Each pair of corresponding elements in the two matrices needs to be added. That means there are \(n^2\) addition operations to be done.

Calculating the total time

If each addition operation takes \(t_{e}\) time, then overall, \(n^2\) operations will take \(n^2 \times t_{e}\) time.

Justifying the answer

The total time depends on the size of the matrices. The larger the matrix (having more elements), the more operations will be needed, and hence, more time will be required. Since each addition operation takes a constant amount of time (\(t_{e}\)), the total additions (basic operations) will determine the overall time required. It clearly demonstrates that the time complexity is \(O(n^2)\), pertaining to the size of the inputs (matrix size in this case).

Key concepts

Algorithms

An algorithm can be understood as a step-by-step procedure or formula for solving a problem. It's a sequence of instructions that are executed to get the desired outcome. When it comes to computational tasks such as matrix addition, the algorithm defines how each element of one matrix is to be added to the corresponding element of another matrix. In our exercise, the algorithm for adding two square matrices involves iterating through each element of the matrices and performing the addition operation. The simplicity or complexity of an algorithm often dictates its efficiency and the time it would take to solve a problem using a computer or by hand as in the given exercise.

Time Complexity

Time complexity is a concept in computer science that describes the amount of time an algorithm takes to complete as a function of the length of the input. It's a crucial aspect of understanding algorithms as it provides insight into how scalable an algorithm is. In the context of our exercise, when talking about adding two square matrices, the time complexity is expressed in terms of the number of basic operations, such as the individual number additions, that need to be performed. Since the exercise requires adding corresponding elements in two matrices with a size of \(n \times n\), the time complexity would be \(O(n^2)\), with each addition being counted as a distinct operation.

Basic Operations in Algorithms

The term 'basic operations' references the most fundamental tasks that an algorithm performs, which often includes arithmetic operations like addition, subtraction, multiplication, and division. These operations are considered to be executed in constant time, denoted by \(t_e\). In our matrix addition problem, the basic operation is the addition of two numbers. As we add corresponding elements from each matrix, these numerous additions are the 'basic operations' foundational to the algorithm's execution. Thus, understanding and optimizing these operations can greatly affect the overall performance and efficiency of an algorithm.

Square Matrix Addition

Square matrix addition is a fundamental operation in algebra where two matrices of equal dimensions are added together by adding their corresponding elements. Each element on matrix A is combined with the corresponding element on matrix B. For square matrices of size \(n \times n\), this involves performing \(n^2\) additions to obtain the resulting matrix. The order in which these additions occur doesn't matter — this is because addition is commutative and associative. However, the number of operations is directly proportional to the square of the matrix's dimension, defining our algorithm's time complexity.