Question

Consider the following declaration: int beta[3][3]; What is stored in beta after each of the following statements executes? a. for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = 0; b. for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = i + j; c. for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = i * j; d. for (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = 2 * (i + j) % 4;

Short answer

a. All zeros; b. Row + column sum; c. Row * column product; d. Double sum mod 4.

Step-by-step solution

Initialize beta with zeros

In the first loop, the code sets each element of the array beta to 0.The code snippet is:```cfor (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = 0;```The inner loop executes first for each value of `i` from 0 to 2. Thus, beta becomes:\[\begin{bmatrix}0 & 0 & 0 \0 & 0 & 0 \0 & 0 & 0\end{bmatrix}\]

Sum of indices

In the second loop, each element of beta is assigned the sum of its indices:The code snippet is:```cfor (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = i + j;```For instance, beta[0][0] becomes 0+0=0, beta[0][1] becomes 0+1=1, etc. Thus, beta becomes:\[\begin{bmatrix}0 & 1 & 2 \1 & 2 & 3 \2 & 3 & 4\end{bmatrix}\]

Product of indices

In the third loop, each element of beta is assigned the product of its indices:The code snippet is:```cfor (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = i * j;```For example, beta[0][0] becomes 0*0=0, beta[1][1] becomes 1*1=1, and so on. Thus, beta becomes:\[\begin{bmatrix}0 & 0 & 0 \0 & 1 & 2 \0 & 2 & 4\end{bmatrix}\]

Double the sum modulo 4

In the last loop, the code assigns each element of beta as two times the sum of its indices modulo 4:The code snippet is:```cfor (i = 0; i < 3; i++) for (j = 0; j < 3; j++) beta[i][j] = 2 * (i + j) % 4;```This means beta[0][1] becomes 2*(0+1)%4=2, beta[2][2] becomes 2*(2+2)%4=0, and so on. Therefore, beta becomes:\[\begin{bmatrix}0 & 2 & 0 \2 & 0 & 2 \0 & 2 & 0\end{bmatrix}\]

Key concepts

Arrays

In C++, an array is a powerful data structure that allows you to store multiple values of the same data type together. This is particularly useful when dealing with matrices or lists. The declaration `int beta[3][3]` creates a two-dimensional array with 3 rows and 3 columns, meaning it can hold 9 integer values in total. Arrays in C++ are zero-indexed, which means the index of the first element is 0. This allows you to access elements using indices like `beta[0][0]`, `beta[1][2]`, etc., which makes it easy to manage and manipulate multiple values within the same array.

Arrays provide a simple way to handle large datasets efficiently. You can perform operations on all the elements, or on specific subsets, which is essential in scenarios such as when filling a matrix with values, clearing it, or aggregating values for further computations.

Nested Loops

Nested loops are loops within loops. They are a fundamental part of programming when it comes to working with multi-dimensional arrays like matrices. In the context of the problem, you see loops such as `for (i = 0; i < 3; i++)` and `for (j = 0; j < 3; j++)`. These loops iterate over elements and allow programmers to perform operations on matrices comprehensively.

In nested loops, the outer loop controls how many times the inner loop will execute. In our example, the outer loop (controlled by index `i`) runs three times, once for each row of the matrix, while the inner loop (controlled by index `j`) executes three times for each column of a row. This structure allows you to access and modify every element of a matrix systematically.

Matrix Initialization

Matrix initialization refers to the process of setting an initial value to each element of a matrix. When you first declare an array such as `int beta[3][3]`, the elements do not have a specific value assigned to them yet, so they need initialization. In the first step of the solution, the elements are all set to zero using nested loops, ensuring each location within the matrix contains a known value before further operations.

Proper initialization is crucial for avoiding random or garbage values that can lead to unpredictable behaviors in your program. It’s also beneficial for setting up a matrix to reflect an initial condition, such as all zeros in mathematical operations, or setting all values to a starting point before collecting data.

Modulo Operation

The modulo operation, represented by the symbol `%` in C++, gives the remainder of division between two integers. It is used in the exercise to apply a transformation to the matrix values. For instance, in the loop: `beta[i][j] = 2 * (i + j) % 4;`, each element is calculated based on the sum of its indices, multiplied by 2, and then taken modulo 4.

This operation serves multiple purposes: it limits the value range of outputs (since the remainder will always be less than the divisor), helps in implementing cyclic patterns, and is often used in hashing algorithms. In our context, it’s being used to limit the matrix values to a range from 0 to 3, thereby forming a cyclic repeating pattern of numbers.