Math 2109: Vector, Matrix and Transform

Math first 3 class note

Course Overview & Syllabus Topics

1. Vector analysis 2. Fourier series, integral, transform (Arif sir) 3. Laplace transformation 4. Matrices (Arif sir)

Recommended Books:

1. Linear Algebra with Applications — Howard Anton
2. Linear Algebra with Applications — David C. Lay

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1st Cycle | 13.07.2026 | Class-01

H.W. Importance of Matrix (Applications & Real-World Usage)

Answer: A Matrix {a two-dimensional rectangular array of numbers or functions arranged in rows and columns} is a fundamental mathematical construct used to represent linear relationships, transform coordinates, and process large multi-variable datasets efficiently.

Key Applications by Domain:

1. Systems of Linear Equations:
   • Provides systematic algorithms (e.g., Gaussian elimination, Matrix Inversion, Cramer’s Rule) to solve simultaneous linear equations rapidly in physics, economics, and structural engineering.
2. Geometric & Spatial Transformations:
   • Represents linear transformations such as 2D/3D rotation, scaling, translation, shear, and reflection using transformation matrices ($T\cdot \vec{v}={\vec{v}}^{\prime }$).
3. Computer Graphics & Game Development:
   • Essential for rendering 3D environments, camera view projections, character animation rigs, and ray tracing in graphics engines (OpenGL, DirectX).
4. Image Processing & Computer Vision:
   • Digital images are stored as matrices where each entry represents pixel brightness/color. Operations like sharpening, blurring, edge detection (Sobel filters), and resizing are executed via matrix convolution.
   • RGB Color Space: Color images are represented by $m\times n\times 3$ tensors (three matrices representing Red, Green, and Blue intensity channels).
5. Digital Signal Processing (DSP) & Communications:
   • Used for Fast Fourier Transforms (FFT), digital filtering, noise removal, data compression (JPEG, MP3), and MIMO antenna channel modeling.
6. Engineering Applications:
   • Electrical: Solving complex AC/DC mesh circuit equations via nodal analysis.
   • Civil & Mechanical: Structural stress analysis, finite element analysis (FEA), and mechanical vibration analysis.
   • Robotics: Kinematic chains and joint positioning using homogeneous transformation matrices.
7. Artificial Intelligence & Machine Learning:
   • Modern deep neural networks represent weights and feature vectors as large matrices; network training relies on massive parallel matrix multiplications (BLAS/CUDA).

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Definition of Matrix

Formal Definition: Matrix

A matrix $A$ of order (or size) $m\times n$ (read as ”$m$ by $n$”) is a rectangular array of $m\cdot n$ numbers (real or complex) arranged in $m$ horizontal rows and $n$ vertical columns, enclosed in square brackets $\left[\begin{array}{c}\dots \end{array}\right]$ or parentheses $(\dots )$.

Mathematically, it is written as:

a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} $$ * **Element Indexing:** $a_{ij}$ denotes the entry located at the **$i$-th row** and **$j$-th column** ($1 \le i \le m$, $1 \le j \le n$). * **Order / Dimension:** $m \times n$, where $m$ is row count and $n$ is column count.

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Types of Matrices (Definitions, Properties & Examples)

1. Row Matrix

A matrix having only one row ($m=1$) and any number of columns ($1\times n$). $A={\left[\begin{array}{ccc}2& 4& 6\end{array}\right]}_{1\times 3}$

2. Column Matrix

A matrix having only one column ($n=1$) and any number of rows ($m\times 1$). $A={\left[\begin{array}{c}2\\ 4\\ 6\end{array}\right]}_{3\times 1}$

3. Rectangular Matrix

A matrix where the number of rows is not equal to the number of columns ($m\ne n$). $A={\left[\begin{array}{ccc}1& 2& 4\\ 3& 5& 6\end{array}\right]}_{2\times 3}$

4. Square Matrix (PYQ: 2017)

A matrix where the number of rows equals the number of columns ($m=n$). A square matrix of size $n\times n$ is said to be of order $n$. $A={\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}_{2\times 2}$

Principal Diagonal & Trace

• Main / Principal Diagonal: Elements $a_{11},a_{22},\dots ,a_{nn}$ where row index equals column index ($i=j$).
• Trace of Matrix $\text{tr}(A)$: The sum of all elements on the principal diagonal: $\text{tr}(A)={\sum }_{i=1}^n{a}_{ii}=a_{11}+a_{22}+\cdots +a_{nn}$.

5. Diagonal Matrix

A square matrix in which all non-diagonal elements are zero, i.e., $a_{ij}=0$ for all $i\ne j$. $A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 7& 0\\ 0& 0& 5\end{array}\right]=\text{diag}(3,7,5)$

6. Scalar Matrix

A diagonal matrix in which all principal diagonal elements are equal to a constant scalar $k$, i.e., $a_{ij}=k$ when $i=j$, and $a_{ij}=0$ when $i\ne j$. $A=\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

7. Identity (Unit) Matrix

A scalar matrix whose principal diagonal elements are all equal to $1$. Denoted by $I$ or $I_n$. $I_2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],I_3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Property of Identity Matrix

For any matrix $A_{m\times n}$: $A\cdot I_n=A$ and $I_m\cdot A=A$.

8. Null (Zero) Matrix

A matrix of any dimension whose entries are all zero. Denoted by $O$ or $O_{m\times n}$. $O=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

9. Symmetric Matrix (PYQ: 2017, 2019)

A square matrix that is equal to its transpose, i.e., $A=A^T$. In element form: $a_{ij}=a_{ji}$ for all $i,j$. $A=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ (Notice mirror symmetry across the principal diagonal: $a_{12}=h=a_{21}$, $a_{13}=g=a_{31}$, $a_{23}=f=a_{32}$).

10. Skew-Symmetric (Anti-Symmetric) Matrix (PYQ: 2015, 2016, 2017, 2019)

A square matrix whose transpose equals its negative, i.e., $A=-A^T$. In element form: $a_{ij}=-a_{ji}$ for all $i,j$. $A=\left[\begin{array}{ccc}0& 2& 3\\ -2& 0& -1\\ -3& 1& 0\end{array}\right]$

Key Exam Proof: Why Diagonal Elements of Skew-Symmetric Matrix Are Zero

For diagonal elements, row index equals column index ($i=j$). By definition of a skew-symmetric matrix: $a_{ii}=-a_{ii}⟹a_{ii}+a_{ii}=0⟹2a_{ii}=0⟹{\mathbf{a}}_{\mathbf{ii}}=0$ Thus, all diagonal entries of any skew-symmetric matrix must always be zero.

11. Upper and Lower Triangular Matrix (PYQ: 2015)

• Upper Triangular Matrix: A square matrix where all entries below the principal diagonal are zero, i.e., $a_{ij}=0$ for all $i>j$. $U=\left[\begin{array}{ccc}2& 3& 5\\ 0& 1& 4\\ 0& 0& 6\end{array}\right]$
• Lower Triangular Matrix: A square matrix where all entries above the principal diagonal are zero, i.e., $a_{ij}=0$ for all $i<j$. $L=\left[\begin{array}{ccc}4& 0& 0\\ 1& 3& 0\\ 7& 2& 8\end{array}\right]$

Determinant Shortcut for Triangular Matrices

The determinant of any upper or lower triangular matrix equals the product of its diagonal elements: $\det (A)=a_{11}\cdot a_{22}\cdots a_{nn}$.

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14.07.26 | Class-02

Key Theorems on Real Matrices

Theorem: Unique Deconstruction of Square Matrices (PYQ: 2017)

Every square matrix $A$ over real numbers can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix.

$\text{Square Matrix }A=\underset{B}{\underbrace{\text{(Symmetric Matrix)}}}+\underset{C}{\underbrace{\text{(Skew-Symmetric Matrix)}}}$

Step-by-Step Proof & Uniqueness Demonstration

Step 1: Algebraic Representation

Let $A$ be any square matrix. We construct two candidate matrices $B$ and $C$: $A=\frac{1}{2}(A+A^T)+\frac{1}{2}(A-A^T)=B+C$ where $B=\frac{1}{2}(A+A^T)$ and $C=\frac{1}{2}(A-A^T)$.

Step 2: Verification of Properties

• Testing $B$ for Symmetric property ($B^T=B$): $B^T={\left[\frac{1}{2}(A+A^T)\right]}^T=\frac{1}{2}\left(A^T+(A^T{)}^T\right)$ Since $(A^T{)}^T=A$: $B^T=\frac{1}{2}(A^T+A)=\frac{1}{2}(A+A^T)=B⟹\mathbf{B}\text{ is symmetric.}$

• Testing $C$ for Skew-Symmetric property ($C^T=-C$): $C^T={\left[\frac{1}{2}(A-A^T)\right]}^T=\frac{1}{2}\left(A^T-(A^T{)}^T\right)$ $C^T=\frac{1}{2}(A^T-A)=-\frac{1}{2}(A-A^T)=-C⟹\mathbf{C}\text{ is skew-symmetric.}$

Step 3: Proof of Uniqueness (Method of Contradiction)

Assume there exists another representation $A=P+Q$, where $P$ is symmetric ($P^T=P$) and $Q$ is skew-symmetric ($Q^T=-Q$). --- (Equation 1)

Taking the transpose of both sides of Equation 1: $A^T=(P+Q{)}^T=P^T+Q^T=P-Q\text{— (Equation 2)}$

• Adding Equation (1) and Equation (2): $A+A^T=(P+Q)+(P-Q)=2P⟹P=\frac{1}{2}(A+A^T)=B$
• Subtracting Equation (2) from Equation (1): $A-A^T=(P+Q)-(P-Q)=2Q⟹Q=\frac{1}{2}(A-A^T)=C$

Since $P=B$ and $Q=C$, the decomposition $A=B+C$ is strictly unique. $■$

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Complex Matrices & Conjugate Transpose

When matrix entries belong to the complex field $\mathbb{C}$ (i.e., $a_{ij}=x+iy$), we define:

1. Complex Conjugate Matrix ($\bar{A}$): Replacing every entry $a_{ij}$ with its complex conjugate ${\bar{a}}_{ij}=x-iy$.
2. Conjugate Transpose Matrix ($A^{\ast }$ or $A^{†}$ or $A^H$): Transpose of the conjugate matrix, i.e., $A^{\ast }=(\bar{A}{)}^T=\overline{(A^T)}$.

Hermitian Matrix (PYQ: 2016, 2017, 2018, 2023, 2025)

A square matrix $A=[a_{ij}]$ over complex numbers is Hermitian if it equals its conjugate transpose: $A^{\ast }=A⟺a_{ij}={\bar{a}}_{ji}\text{for all }i,j$ $A=\left[\begin{array}{ccc}2& 1+i& 2-i\\ 1-i& 3& -1-3i\\ 2+i& -1+3i& 4\end{array}\right]$

Key Exam Property: Diagonal Elements of Hermitian Matrix Are Real $i=j$), we have $a_{ii}={\bar{a}}_{ii}$. If $a_{ii}=x+iy$, then $x+iy=x-iy⟹2iy=0⟹y=0$. Thus, all principal diagonal elements of a Hermitian matrix MUST be purely real numbers.

For diagonal elements (

Skew-Hermitian Matrix (PYQ: 2015, 2016, 2017, 2018, 2021, 2025)

A square complex matrix $A$ is Skew-Hermitian if its conjugate transpose equals its negative: $A^{\ast }=-A⟺a_{ij}=-{\bar{a}}_{ji}\text{for all }i,j$ $A=\left[\begin{array}{ccc}0& 1+i& 3+2i\\ -1+i& 0& i\\ -3+2i& i& 0\end{array}\right]$

Key Exam Property: Diagonal Elements of Skew-Hermitian Matrix $i=j$), we have $a_{ii}=-{\bar{a}}_{ii}⟹a_{ii}+{\bar{a}}_{ii}=0$. If $a_{ii}=x+iy$, then $(x+iy)+(x-iy)=2x=0⟹x=0$. Thus, all principal diagonal elements of a Skew-Hermitian matrix MUST be either zero or purely imaginary.

For diagonal elements (

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H.W. / Proof: Complex Square Matrix = Hermitian + Skew-Hermitian (PYQ: 2018)

Any square matrix $A$ with complex entries can be uniquely expressed as $A=B+C$, where $B$ is Hermitian ($B^{\ast }=B$) and $C$ is Skew-Hermitian ($C^{\ast }=-C$).

Proof: Write $A$ as: $A=\underset{B}{\underbrace{\frac{1}{2}(A+A^{\ast })}}+\underset{C}{\underbrace{\frac{1}{2}(A-A^{\ast })}}$

• Testing $B$ for Hermitian property: $B^{\ast }={\left[\frac{1}{2}(A+A^{\ast })\right]}^{\ast }=\frac{1}{2}\left(A^{\ast }+(A^{\ast }{)}^{\ast }\right)$ Since $(A^{\ast }{)}^{\ast }=A$: $B^{\ast }=\frac{1}{2}(A^{\ast }+A)=B⟹\mathbf{B}\text{ is Hermitian.}$

• Testing $C$ for Skew-Hermitian property: $C^{\ast }={\left[\frac{1}{2}(A-A^{\ast })\right]}^{\ast }=\frac{1}{2}\left(A^{\ast }-(A^{\ast }{)}^{\ast }\right)$ $C^{\ast }=\frac{1}{2}(A^{\ast }-A)=-\frac{1}{2}(A-A^{\ast })=-C⟹\mathbf{C}\text{ is Skew-Hermitian.}■$

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Special Types of Square Matrices (Summary Table & Definitions)

Matrix Type	Defining Condition	Key Properties / Determinant / Inverses	Exam PYQ Tag
Singular	$\det (A)=\Vert A\Vert =0$	Has no matrix inverse ($A^{-1}$ does not exist).	PYQ: 2017
Non-Singular	$\det (A)=\Vert A\Vert \ne 0$	Inverse $A^{-1}$ exists.	-
Orthogonal	$A{A}^T=A^{T}A=I$	$A^{-1}=A^T$, $\det (A)=\pm 1$.	PYQ: 2015
Idempotent	$A^2=A$	$A^k=A$ for any integer $k\ge 1$. Eigenvalues are $0$ or $1$.	Class Concept
Nilpotent	$A^n=O,A^{n-1}\ne O$	$n$ is index/order of nilpotency. $\det (A)=0$.	PYQ: 2015
Periodic	$A^{m+1}=A$	$m$ is the period (Idempotent has period $m=1$).	Class Concept
Involutory	$A^2=I$	$A^{-1}=A$, $\det (A)=\pm 1$.	Class Concept
Unitary	$A{A}^{\ast }=A^{\ast }A=I$	Complex analogue of Orthogonal matrix ($A^{-1}=A^{\ast }$). $\Vert \det (A)\Vert =1$.	PYQ: 2023

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Class Solved Problem: Orthogonal Matrix Parameter Determination

Solved Problem: Finding Unknown Parameters for Orthogonality

Question: Determine the values of $\alpha ,\beta ,\gamma$ such that the matrix $A$ is orthogonal: $A=\left[\begin{array}{ccc}0& 2\beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma \end{array}\right]$

Solution: By definition of an orthogonal matrix, $A{A}^T=I_3$.

First, write the transpose $A^T$: $A^T=\left[\begin{array}{ccc}0& \alpha & \alpha \\ 2\beta & \beta & -\beta \\ \gamma & -\gamma & \gamma \end{array}\right]$

Multiply $A$ and $A^T$: $A{A}^T=\left[\begin{array}{ccc}0& 2\beta & \gamma \\ \alpha & \beta & -\gamma \\ \alpha & -\beta & \gamma \end{array}\right]\left[\begin{array}{ccc}0& \alpha & \alpha \\ 2\beta & \beta & -\beta \\ \gamma & -\gamma & \gamma \end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$

Computing individual entry products:

4\beta^2 + \gamma^2 & 2\beta^2 - \gamma^2 & -2\beta^2 + \gamma^2 \\ 2\beta^2 - \gamma^2 & \alpha^2 + \beta^2 + \gamma^2 & \alpha^2 - \beta^2 - \gamma^2 \\ -2\beta^2 + \gamma^2 & \alpha^2 - \beta^2 - \gamma^2 & \alpha^2 + \beta^2 + \gamma^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ Equating corresponding elements: 1. From entry $(1, 2)$: $$ 2\beta^2 - \gamma^2 = 0 \implies \gamma^2 = 2\beta^2 \quad \text{--- (1)} $$ 2. From entry $(2, 3)$: $$ \alpha^2 - \beta^2 - \gamma^2 = 0 \implies \alpha^2 = \beta^2 + \gamma^2 \quad \text{--- (2)} $$ Substitute (1) into (2): $$ \alpha^2 = \beta^2 + 2\beta^2 \implies \alpha^2 = 3\beta^2 \quad \text{--- (3)} $$ 3. From diagonal entry $(2, 2)$: $$ \alpha^2 + \beta^2 + \gamma^2 = 1 \quad \text{--- (4)} $$ Substitute (1) and (3) into (4): $$ 3\beta^2 + \beta^2 + 2\beta^2 = 1 \implies 6\beta^2 = 1 \implies \mathbf{\beta = \pm \frac{1}{\sqrt{6}}} $$ Now calculate $\gamma$ and $\alpha$: * Using (1): $\gamma^2 = 2\left(\frac{1}{6}\right) = \frac{1}{3} \implies \mathbf{\gamma = \pm \frac{1}{\sqrt{3}}}$ * Using (3): $\alpha^2 = 3\left(\frac{1}{6}\right) = \frac{1}{2} \implies \mathbf{\alpha = \pm \frac{1}{\sqrt{2}}}$ **Final Answer:** $$ \mathbf{\alpha = \pm \frac{1}{\sqrt{2}}, \quad \beta = \pm \frac{1}{\sqrt{6}}, \quad \gamma = \pm \frac{1}{\sqrt{3}}} $$

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Partitioning & Block Matrices

Matrix partitioning involves dividing a large matrix into smaller sub-blocks by horizontal and vertical lines.

Partitioning Rule for Matrix Multiplication

Two partitioned matrices $A$ and $B$ are conformable for block multiplication if the column partitioning of $A$ matches the row partitioning of $B$.

Worked Example: Block Multiplication

Consider partitioned matrices:

B = \left[ \begin{array}{c|cc} 3 & 1 & 2 \\ 5 & 2 & 6 \\ \hline 4 & 7 & 8 \end{array} \right] = \left[ \begin{array}{c|c} B_{11} & B_{12} \\ \hline B_{21} & B_{22} \end{array} \right] $$ **Sub-block Definitions:** $$ A_{11} = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \quad A_{12} = \begin{bmatrix} 4 \\ 5 \end{bmatrix}, \quad A_{21} = \begin{bmatrix} 6 & 2 \\ 1 & 2 \end{bmatrix}, \quad A_{22} = \begin{bmatrix} 4 \\ 4 \end{bmatrix} $$ $$ B_{11} = \begin{bmatrix} 3 \\ 5 \end{bmatrix}, \quad B_{12} = \begin{bmatrix} 1 & 2 \\ 2 & 6 \end{bmatrix}, \quad B_{21} = \begin{bmatrix} 4 \end{bmatrix}, \quad B_{22} = \begin{bmatrix} 7 & 8 \end{bmatrix} $$ **Block Product Expression:** $$ A B = \left[ \begin{array}{c|c} A_{11} B_{11} + A_{12} B_{21} & A_{11} B_{12} + A_{12} B_{22} \\ \hline A_{21} B_{11} + A_{22} B_{21} & A_{21} B_{12} + A_{22} B_{22} \end{array} \right] $$ **Block Computation Steps:** 1. Top-Left Block: $A_{11}B_{11} + A_{12}B_{21} = \begin{bmatrix} 1(3)+2(5) \\ 0(3)+1(5) \end{bmatrix} + \begin{bmatrix} 4(4) \\ 5(4) \end{bmatrix} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} + \begin{bmatrix} 16 \\ 20 \end{bmatrix} = \begin{bmatrix} 29 \\ 25 \end{bmatrix}$ 2. Top-Right Block: $A_{11}B_{12} + A_{12}B_{22} = \begin{bmatrix} 5 & 14 \\ 2 & 6 \end{bmatrix} + \begin{bmatrix} 28 & 32 \\ 35 & 40 \end{bmatrix} = \begin{bmatrix} 33 & 46 \\ 37 & 46 \end{bmatrix}$ 3. Bottom-Left Block: $A_{21}B_{11} + A_{22}B_{21} = \begin{bmatrix} 28 \\ 13 \end{bmatrix} + \begin{bmatrix} 16 \\ 16 \end{bmatrix} = \begin{bmatrix} 44 \\ 29 \end{bmatrix}$ 4. Bottom-Right Block: $A_{21}B_{12} + A_{22}B_{22} = \begin{bmatrix} 10 & 24 \\ 5 & 14 \end{bmatrix} + \begin{bmatrix} 28 & 32 \\ 28 & 32 \end{bmatrix} = \begin{bmatrix} 38 & 56 \\ 33 & 46 \end{bmatrix}$ $$ \therefore AB = \begin{bmatrix} 29 & 33 & 46 \\ 25 & 37 & 46 \\ 44 & 38 & 56 \\ 29 & 33 & 46 \end{bmatrix} $$ *(Note: Preserving conceptual block structure from class notes while explicitly validating exact linear combinations).* --- ## 2nd Cycle | 20.07.26 | Class-03 ### Elementary Transformations (Elementary Row Operations) *(PYQ: 2016, 2023)* An **Elementary Transformation** refers to performing any of three fundamental operations on rows (or columns) of a matrix without changing its solution space or rank. > [!abstract] Three Elementary Row Operations (EROs) > > 1. **Row Interchange ($R_i \leftrightarrow R_j$):** Swap two rows $i$ and $j$. > 2. **Row Scaling ($R_i \rightarrow k R_i$):** Multiply all entries in row $i$ by a non-zero constant scalar $k \neq 0$. > 3. **Row Addition ($R_i \rightarrow R_i + k R_j$):** Add to row $i$ a scalar multiple $k$ of row $j$. > [!info] Matrix Equivalence ($\sim$) > > If a matrix $B$ is obtained from $A$ by performing a finite sequence of elementary row operations, then $A$ and $B$ are **equivalent matrices**, denoted as: > $$ A \sim B $$ > *(Note: Equivalent matrices are not equal $A \neq B$, but they share identical rank, determinant properties, and system solutions)*. #### Elementary Matrix Definition An **Elementary Matrix** $E$ is a matrix obtained by performing a **single** elementary row operation on an Identity Matrix $I$. * *Property:* Applying an ERO to matrix $A$ is mathematically equivalent to pre-multiplying $A$ by the corresponding elementary matrix $E$: $A \to E \cdot A$. #### Examples of Elementary Transformations: Given matrix $A = \begin{bmatrix} 3 & 1 & 2 \\ 5 & 2 & 0 \\ 1 & 2 & 4 \end{bmatrix}$: * **Step 1 ($R_1 \leftrightarrow R_3$):** Swap row 1 and row 3 to get a leading $1$ at $a_{11}$: $$ \begin{bmatrix} 3 & 1 & 2 \\ 5 & 2 & 0 \\ 1 & 2 & 4 \end{bmatrix} \sim \begin{bmatrix} 1 & 2 & 4 \\ 5 & 2 & 0 \\ 3 & 1 & 2 \end{bmatrix} $$ * **Step 2 ($R_2 \rightarrow \frac{1}{5} R_2$):** Scale row 2 by $\frac{1}{5}$: $$ \sim \begin{bmatrix} 1 & 2 & 4 \\ 1 & 2/5 & 0 \\ 3 & 1 & 2 \end{bmatrix} $$ * **Step 3 ($R_2 \rightarrow R_2 - R_1$):** Subtract row 1 from row 2: $$ \sim \begin{bmatrix} 1 & 2 & 4 \\ 0 & -8/5 & -4 \\ 3 & 1 & 2 \end{bmatrix} $$ --- > [!warning] Key Exam Checkpoint: Strict Rules for Writing Row Operations > > To avoid losing marks in exams, follow these exact syntax rules: > 1. **Target Row Order:** In $R_i' \rightarrow R_i \pm k R_j$, the row being modified ($R_i$) **MUST be written first**. > * $\checkmark \quad R_2' \rightarrow R_2 - 3R_1$ (Correct) > * $\mathbf{\times} \quad R_2' \rightarrow 3R_1 - R_2$ (Incorrect notation format in rigorous exams) > 2. **No Simultaneous Inter-dependency Bug:** Never alter two rows based on each other in the exact same step! > * $\mathbf{\times} \quad \begin{cases} R_1' \rightarrow R_1 - R_2 \\ R_2' \rightarrow R_2 - R_1 \end{cases}$ (Will destroy information and zero out rows invalidly). --- ### Rank of a Matrix *(PYQ: 2019, 2025)* > [!abstract] Dual Definition of Matrix Rank > > > **Definition 1 (Linear Independence):** > The **rank** of a matrix $A$, denoted by $\text{rank}(A)$ or $\rho(A)$, is the maximum number of **linearly independent rows** (or columns) in $A$. > > **Definition 2 (Minor-Based Formal Definition):** > A matrix $A$ is said to have rank $r$ if and only if: > 1. There exists at least **one non-zero minor** of order $r$ in $A$. > 2. **Every minor** of order higher than $r$ (i.e., order $r+1$ or greater) vanishes completely (equals zero). > [!tip] Essential Properties of Matrix Rank > > 1. For any matrix $A_{m \times n}$: $\text{rank}(A) \le \min(m, n)$. > 2. $\text{rank}(A) = \text{rank}(A^T)$. > 3. Elementary row transformations **do not alter** the rank of a matrix. > 4. $\text{rank}(A B) \le \min(\text{rank}(A), \text{rank}(B))$. > 5. A square matrix $A_{n \times n}$ is non-singular $\iff \text{rank}(A) = n$. --- ### Row-Echelon Form (REF) & Reduced Row-Echelon Form (RREF) A matrix is in **Row-Echelon Form (REF)** if it satisfies three structural conditions: > [!abstract] Conditions for Row-Echelon Form (REF) > > 1. **Zero Rows at Bottom:** All rows consisting entirely of zeros are grouped at the bottom of the matrix. > 2. **Staircase Leading Entries:** The leading entry *{the first non-zero number from the left}* of a non-zero row is strictly to the right of the leading entry of the row above it. > 3. **Zeros Below Pivots:** All entries in a column directly below a leading entry are zero. #### Examples of Row-Echelon Form: $$ A = \begin{bmatrix} \mathbf{3} & 2 & 4 \\ 0 & \mathbf{1} & 2 \\ 0 & 0 & 0 \end{bmatrix} $$ *(Leading entries / pivots are $\mathbf{3}$ and $\mathbf{1}$. Notice all entries below pivots are zero)*. $$ B = \begin{bmatrix} 0 & \mathbf{3} & 2 & 1 \\ 0 & 0 & \mathbf{2} & 4 \\ 0 & 0 & 0 & 0 \end{bmatrix} $$ > [!note] How to Find Rank Using Row-Echelon Form > > Once a matrix $A$ is transformed into Row-Echelon Form via elementary row operations: > $$ \mathbf{\text{rank}(A) = \text{Number of Non-Zero Rows in its Row-Echelon Form}} $$ --- > [!info] Extension: Reduced Row-Echelon Form (RREF) > > A matrix is in **Reduced Row-Echelon Form (RREF)** if it satisfies all REF conditions PLUS: > 1. Every leading entry (pivot) is strictly **$1$**. > 2. Every pivot $1$ is the **only non-zero entry in its entire column** (entries both above and below the pivot are zero). ~ Nakib :)