quiz.json

{
  "questions": [
    {
      "stage": "pre",
      "question": "For matrix multiplication (m x n) @ (n x p), what must be true about the dimensions?",
      "options": ["m must equal p", "The inner dimensions n must match", "All dimensions must be equal", "m must be greater than p"],
      "correct": 1,
      "explanation": "Matrix multiplication requires the number of columns in the first matrix (n) to equal the number of rows in the second matrix (n). The result has shape (m x p)."
    },
    {
      "stage": "pre",
      "question": "What is the identity matrix?",
      "options": ["A matrix of all ones", "A square matrix with ones on the diagonal and zeros elsewhere that acts as the multiplicative identity", "A matrix where every element is unique", "The transpose of any given matrix"],
      "correct": 1,
      "explanation": "The identity matrix I has ones on the diagonal and zeros everywhere else. Multiplying any matrix by I returns the original matrix unchanged, like multiplying a number by 1."
    },
    {
      "stage": "post",
      "question": "What is the key difference between element-wise multiplication and matrix multiplication?",
      "options": ["Element-wise is faster while matrix multiplication is more accurate", "Element-wise multiplies matching positions (same shape required), matrix multiplication takes dot products of rows and columns (inner dimensions must match)", "They produce the same result but use different notation", "Element-wise only works on vectors while matrix multiplication works on matrices"],
      "correct": 1,
      "explanation": "Element-wise (Hadamard) product multiplies corresponding elements and requires identical shapes. Matrix multiplication computes dot products between rows and columns with the rule (m,n)@(n,p)=(m,p)."
    },
    {
      "stage": "post",
      "question": "In the expression 'output = relu(W @ x + b)', what role does broadcasting play?",
      "options": ["It broadcasts the computation across multiple GPUs", "It automatically stretches the bias vector b to match the shape of W @ x so they can be added", "It converts the data types of W and x to match", "It repeats the relu activation across all elements"],
      "correct": 1,
      "explanation": "W @ x produces a column vector, and b is also a vector. Broadcasting stretches b across the batch dimension if needed, allowing element-wise addition without explicit shape matching."
    },
    {
      "stage": "post",
      "question": "What does a determinant of zero for a matrix indicate?",
      "options": ["The matrix has all zero entries", "The matrix is singular: it crushes at least one dimension, cannot be inverted, and has no unique solution", "The matrix is the identity matrix", "The matrix performs a rotation"],
      "correct": 1,
      "explanation": "A zero determinant means the transformation collapses space by at least one dimension (e.g., mapping 2D to a line). The matrix has no inverse, and linear systems using it have either no solution or infinitely many."
    }
  ]
}