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Mathematical Methods Chapter 3 Ex 3.1 Solved Notes & Quiz
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Mathematical Methods Chapter 3 Ex 3.1 Solved Notes & Quiz
Welcome to the authoritative study guide for S.M. Yusuf Mathematical Methods Chapter 3 Exercise 3.1, curated specifically for BS Mathematics, ADP, and University students across Punjab University (PU), University of Sargodha (UOS), Quaid-i-Azam University (QAU), and affiliated institutions in Pakistan. In modern linear algebra, mastering matrix transformations, elementary row operations, vector dependence, and system solver algorithms forms the bedrock of advanced abstract algebra and computational mathematics.
Before attempting the exercise problems, students must be thoroughly familiar with the underlying matrix algebra theorems, invertibility conditions, and echelon reduction definitions summarized in the canonical table below.
| Concept / Theorem | Mathematical Definition / Formula | Key Application |
|---|---|---|
| Symmetric & Skew-Symmetric Decomposition | $A = S + K$, where $S = \frac{1}{2}(A + A^T)$ and $K = \frac{1}{2}(A - A^T)$ | Decomposing any square matrix into symmetric ($S^T=S$) and skew-symmetric ($K^T=-K$) components. |
| Matrix Inverse via Adjoint | $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, where $\text{adj}(A) = (C_{ij})^T$ | Solving non-singular linear systems $AX = B \implies X = A^{-1}B$. |
| Row Echelon Form (REF) | 1. Zero rows are at the bottom. 2. Leading entry of a row is strictly to the right of the leading entry above it. |
Gaussian elimination and computing matrix rank ($\text{Rank}(A) = \text{number of non-zero rows}$). |
| Reduced Row Echelon Form (RREF) | Matrix is in REF, every leading entry is $1$, and each leading $1$ is the only non-zero entry in its column. | Gauss-Jordan elimination and direct computation of unique solutions or free parameters. |
| Linear Dependence Test | $c_1 v_1 + c_2 v_2 + \dots + c_n v_n = \mathbf{0}$ has non-zero constants $c_i \iff \det(V) = 0$. | Determining vector space bases and linear independence of vectors in $\mathbb{R}^n$. |
Question 1 (Part i): Express the matrix $A = \begin{bmatrix} 2 & 4 & -1 \\ 0 & 3 & 5 \\ 1 & -2 & 6 \end{bmatrix}$ as the sum of a symmetric matrix $S$ and a skew-symmetric matrix $K$.
Solution:
First, write the transpose of matrix $A$, denoted as $A^T$:
$$A^T = \begin{bmatrix} 2 & 0 & 1 \\ 4 & 3 & -2 \\ -1 & 5 & 6 \end{bmatrix}$$
The symmetric component $S$ is defined as $S = \frac