Mathematics 2 Week 1 Graded Assignment Solution

🔥 Mathematics 2 Week 1 Graded Assignment

Versions: v2_t22024 (10 Questions), v1_t12023 (15 Questions) — The v1_t12023 version is the most comprehensive set with 15 problems.

This downloadable PDF bundle provides complete, step-by-step solutions for Week 1 of “Mathematics for Data Science-2.” This foundational module, Vectors and Matrices, is crucial for understanding the linear algebra basis of Machine Learning and Data Science. Below is a detailed, problem-by-problem breakdown of the two major versions, ensuring you know exactly what topics are covered:

Version v1_t12023 (15 Problems) – Comprehensive Set (Recommended)

• Q1: Matrix Classification (MSQ): Match four 3×3 matrices (e.g., lower triangular, scalar, diagonal) to their correct properties, including determining if a matrix has det(A) = 0.
• Q2: SLE Matching (MSQ): Match three systems of linear equations (SLE) with their solution type (Unique, Infinite, No Solution) and geometric representations (e.g., intersecting lines, parallel planes).
• Q3: Matrix Order & Existence (MSQ): Given B (3×2) and C (2×2), determine the required order of matrix A in AB = C and prove there are infinitely many such matrices.
• Q4: Characteristic Polynomial (MSQ): For a 2×2 matrix A, derive the full formula for det(A − cI) as a polynomial in terms of trace(A) and det(A).
• Q5: Real-world SLE (MSQ): Formulate the matrix representation for fruit purchase costs and identify the correct solution properties (e.g., x = 2, y = 3 is a solution; infinitely many real solutions exist).
• Q6: Determinant Property (NAT): Find n where det(3A) = n × det(A) for a 3×3 matrix A.
• Q7: Determinant (NAT): Calculate det(A) for a 3×3 matrix with column dependence (e.g., C₃ = C₁ + 10C₂), showing det(A) = 0.
• Q8: Matrix Algebra (NAT): Given A² = A, find scalar m in the identity (I + A)³ − A = I + mA.
• Q9: Determinant (NAT): Calculate det(A) for a 3×3 matrix where rows are linearly dependent.
• Q10: Matrix Product Determinant (NAT): Find det(2A²B⁻¹) for a 3×3 matrix A, where det(A) = 1 and B is derived from A using row operations.
• Q11: Matrix Power and Trace (NAT): Given A is an all-ones 3×3 matrix, find the sum of the diagonal elements (Trace) of A⁴.
• Q12: Generated Matrix Determinant (NAT): Find det(A) for a 3×3 matrix where αᵢⱼ = i + j, proving det(A) = 0 using row operations.
• Q13–15: Comprehension – Electricity Cost (MSQ/NAT): Model the system Ax = b based on monthly utility consumption and payments, find the coefficient matrix A, solve for the sum of per-unit rates (Σx′), and identify the correct determinant expansion for det(A).

Version v2_t22024 (10 Problems) – Focused Set

• Q1: SLE Matching (MSQ): Match systems of linear equations with solution type and geometric representations (e.g., parallel planes).
• Q2: Determinant Property (NAT): Find n where det(4A) = n × det(A) for a 5×5 matrix A, using 4⁵.
• Q3: Matrix Algebra (NAT): Given A² = A, find scalar m in the identity (I + A)³ − 10A = I + mA.
• Q4: Determinant (NAT): Calculate det(A) for a 3×3 matrix with row/column dependency.
• Q5: Matrix Product Determinant (NAT): Find det(4A²B⁻¹) for a 3×3 matrix A (det(A) = 5), where B is derived from A using row operations.
• Q6: Matrix Power and Trace (NAT): Given A is an all-ones 3×3 matrix, find the sum of the diagonal elements (Trace) of A⁷.
• Q7: Generated Matrix Determinant (NAT): Find det(A) for a 3×3 matrix where aᵢⱼ = 1i + 3j.
• Q8–10: Comprehension – Weighted Scores (MSQ/NAT): Model student marks (m, p, c) using weighted averages, identify the coefficient matrix A, solve for average score (m + p + c)/3 = 80, and calculate det(A).

File Format & Compatibility:

• File Type: PDF (Portable Document Format)
• v1_t12023: Approx. 6 MB (16 pages)
• v2_t22024: Approx. 4 MB (8 pages)
• v3_t32025: Approx. 3 MB (8 pages)
• Step-by-step solutions with full mathematical notation (LaTeX-style)
• Optimized for the IIT Madras BS Degree program (Foundation/Diploma Level)