Q1. You are a data scientist analyzing customer data for an e-commerce site. You have 1000 customers and 50 features (age, purchase history, etc.). How would you represent this data in a structured way?

This data is naturally represented as a matrix with 1000 rows (customers) and 50 columns (features).
Each entry is the value of a feature for a customer.
Matrices allow you to perform operations like scaling, transforming, or applying machine learning algorithms (e.g., linear regression: y = X·w).
Libraries like NumPy and pandas store data as matrices for efficient computation.

Q2. In image processing, a grayscale image is a matrix of pixel intensities (0-255). If you have a 100x100 image and you want to blur it using a 3x3 averaging kernel, how would you apply this kernel using matrix operations?

Convolution: Slide the kernel over the image matrix.
For each position, multiply the kernel element-wise with the overlapping image patch and sum.
This yields a new matrix of the same size (with padding).
Matrix convolution is the basis of edge detection, blurring, and feature extraction in neural networks (CNNs).

Q3. A social network has users as rows and columns, and matrix entry (i,j) = 1 if user i follows user j. How would you find the number of mutual follows between users? What operation gives the number of two-step connections?

The matrix is symmetric for mutual follows (M[i,j] = M[j,i] = 1).
Count mutual follows by summing the lower triangle of the matrix.
For two-step connections (follows of follows), multiply the matrix by itself:
M²[i,j] counts the number of paths of length 2 from i to j.
This is a key concept in network analysis (e.g., PageRank algorithm).

Q4. In a recommendation system, you have a user-item matrix where entries are ratings. This matrix is very sparse (most entries empty). How would you find similar users without filling all empty cells?

Use matrix factorization: decompose the matrix into two smaller matrices U (user factors) and V (item factors) such that U·V^T approximates the rating matrix.
Then user similarity can be computed from U.
This is the basis of collaborative filtering algorithms used by Netflix and Amazon.

Q5. You have a matrix of pixel values for a batch of 100 grayscale images, each 28x28 pixels. What is the shape of this matrix if you flatten each image into a vector? How would you store a batch for neural network input?

Flatten each 28×28 image into a 1D vector of length 784.
The batch matrix becomes 100 rows (images) × 784 columns (pixels).
This is the standard input format for fully connected neural networks.
Then you can multiply by weight matrices to compute predictions.