Lesson 1.1: Linear Algebra (Matrices, Rank, Determinants, Eigenvalues)

Linear Algebra forms the backbone of Engineering Mathematics. Topics like Matrices, Rank, Determinants, Eigenvalues and Eigenvectors are frequently tested in GATE and PSU exams because they have direct applications in Engineering Mechanics, Vibrations, Numerical Methods, Structural Analysis and Control Systems.

🔹 1. Matrices

Definition

A Matrix is a rectangular arrangement of numbers into rows and columns.
A matrix of order $m\times n$ has m rows and n columns.

A=[a11a12…a1na21a22…a2n⋮⋮⋱⋮am1am2…amn]A = \begin{bmatrix} 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}A=​a11​a21​⋮am1​​a12​a22​⋮am2​​……⋱…​a1n​a2n​⋮amn​​​

Types of Matrices

• Row Matrix: Only one row

• Column Matrix: Only one column

• Square Matrix: Equal number of rows & columns

• Diagonal Matrix: Non-zero only on diagonal

• Identity Matrix (I): Diagonal elements = 1

• Zero Matrix: All elements = 0

• Symmetric Matrix: A=ATA = A^TA=AT

• Skew-Symmetric Matrix: A=−ATA = -A^TA=−AT

• Orthogonal Matrix: ATA=IA^T A = IATA=I

Operations on Matrices

• Addition/Subtraction: Element-wise

• Multiplication:

  • (AB)≠(BA)(AB) \neq (BA)(AB)=(BA) (Not commutative)

  • Associative & distributive properties hold

• Transpose: Flipping rows into columns

• Inverse:
  A−1=1∣A∣adj(A),∣A∣≠0A^{-1} = \frac{1}{|A|} \text{adj}(A), \quad |A|\neq 0A−1=∣A∣1​adj(A),∣A∣=0

🔹 2. Rank of a Matrix

Definition

The rank of a matrix is the maximum number of linearly independent rows or columns.

Methods to Find Rank

• Echelon Form: Convert matrix to row echelon form (REF/RREF) using Gaussian elimination.

• Minor Method: Highest order of non-zero determinant of sub-matrix.

Consistency of Linear Equations (Important for PSU)

• If $\text{Rank}(A)=\text{Rank}([A|B])=n$ → Unique solution

• If $\text{Rank}(A)=\text{Rank}([A|B])<n$ → Infinite solutions

• If Rank(A)≠Rank([A∣B])\text{Rank}(A) \neq \text{Rank}([A|B])Rank(A)=Rank([A∣B]) → No solution

🔹 3. Determinants

Definition

The determinant of a square matrix is a scalar value obtained from its elements.

Properties of Determinants

1. If two rows/columns are identical → Determinant = 0

2. Interchanging rows/columns → Sign changes

3. Multiplying a row/column by k → Determinant multiplied by k

4. If a row/column is expressed as sum → Determinant can be split

5. If matrix is triangular → Determinant = Product of diagonal elements

Applications

• Solving linear equations using Cramer’s Rule

• Checking linear independence of vectors

• Area/volume representation in geometry

🔹 4. Eigenvalues & Eigenvectors

Definition

For a square matrix $A$, if

$$Ax=\lambda x$$

where x≠0x \neq 0x=0, then:

• $\lambda$ = Eigenvalue

• $x$ = Eigenvector

Finding Eigenvalues

Solve Characteristic Equation:

$$|A-\lambda I|=0$$

Properties

1. Sum of eigenvalues = Trace of A

2. Product of eigenvalues = Determinant of A

3. Eigenvectors corresponding to distinct eigenvalues are linearly independent

4. For symmetric matrices → Eigenvalues are always real

Applications in Engineering

• Vibrations & Modal Analysis

• Stress Analysis (Principal stresses are eigenvalues)

• Stability of systems (control engineering, power systems)

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE 2019 ME):
If A=[2103]A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}A=[20​13​], find its Eigenvalues.
👉 Solution:
$|A-\lambda I|=(2-\lambda )(3-\lambda )=0$
→ λ = 2, 3

Example 2 (GATE 2016 ME):
Find Rank of

A=[1224]A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}A=[12​24​]

👉 Solution: Rows are multiples → Rank = 1

🔹 6. Practice Exercises

1. Find eigenvalues of

A=[4−211]A = \begin{bmatrix} 4 & -2 \\ 1 & 1 \end{bmatrix}A=[41​−21​]

2. Reduce

A=[246123369]A = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 2 & 3 \\ 3 & 6 & 9 \end{bmatrix}A=​213​426​639​​

to REF and find its Rank.
3. Use Cramer’s Rule to solve:

$$x+y+z=6,x-y+z=2,2x+y-z=1$$

🔹 7. Summary

• Matrices: Rectangular arrays, operations, inverse.

• Rank: Number of independent rows/columns → consistency of equations.

• Determinants: Scalar value with properties → used in solving systems.

• Eigenvalues/Eigenvectors: Fundamental in engineering applications like vibrations, stress analysis.