Question

Design a function that takes in an \(\mathrm{M} \times \mathrm{N}\) matrix as a parameter and modifies it such that if an element in it is 0 , its entire row and column are set to 0 .

Short answer

Find zero positions, zero respective rows and columns, and return the modified matrix.

Step-by-step solution

Identify Matrix Dimensions

First, determine the dimensions of the matrix. If it's an \( M \times N \) matrix, \( M \) represents the number of rows and \( N \) represents the number of columns.

Locate Zeros in Matrix

Iterate through each element in the matrix to identify the positions (row and column indices) where zeros are present. Store these indices separately for rows and columns.

Zero Out Identified Rows

Using the indices gathered for rows containing zeros, set all elements in these rows to zero. This ensures that each row with a zero is fully zeroed out.

Zero Out Identified Columns

Similarly, use the indices for columns containing zeros to set all elements in these columns to zero. This completes the zeroing for columns.

Return the Modified Matrix

The function should now return the modified matrix where all corresponding rows and columns of any originally zero element have been set to zero.

Key concepts

Matrix Dimensions

To begin understanding matrix manipulation, it is crucial to first grasp the concept of matrix dimensions. A matrix is an array of numbers laid out in rows and columns. An \( M \times N \) matrix has \( M \) rows and \( N \) columns. It’s like a grid where each cell holds a number. Consider the concept of dimensions as a matrix’s blueprint. Knowing the matrix's dimensions helps you understand its structure and how to interact with it. For example, a \(3 \times 2 \) matrix would have three rows and two columns:

• A total of 3 sets of horizontal entries.
• A total of 2 vertical lines of entries.

By knowing a matrix's dimensions, you can better understand how to navigate through it when performing operations, such as zeroing out elements.

Zeroing Matrix Elements

Zeroing matrix elements is an important concept when working with matrices. This process involves setting specific parts of a matrix to zero based on given conditions. In your matrix modification function, the goal is to zero out entire rows and columns if any of their elements are zero.

Locating Zero Elements

To achieve this, you must first find where the zero elements are within the matrix. The typical approach is to iterate over each element, identifying the positions, or indices, of these zeros.

Applying Zeroing

Once you have identified these indices:

• Ensure every element in the same row as any zero is set to zero.
• Ensure every element in the same column as any zero is also set to zero.

This helps maintain consistency in the matrix following a zero finding. Keeping track of indices and applying changes helps achieve the desired modification without losing track of original zero positions.

Iterating through Matrices

Iterating through matrices involves inspecting each element, often by using two nested loops. The outer loop runs through each row, and the inner loop accesses each column within the current row.

Nested Loop Approach

For a matrix defined as \( M \times N \):

• The outer loop would typically iterate \( M \) times.
• The inner loop iterates \( N \) times.

This ensures that each element, from top to bottom and left to right, is examined and can be processed, such as checking for zeros.

Efficacy in Matrix Processing

Iteration is vital for applying operations like zeroing out rows and columns. By systematically moving through each element, changes can be applied accurately based on the conditions identified. Thus, iterating through a matrix is a foundational step in most matrix operations, allowing effective and logical manipulation.