Linear Algebra and Matrix Operations

Linear algebra provides the computational tools for solving simultaneous equations — a skill used in statics, circuit analysis, structural analysis, and virtually every engineering discipline.

Matrix Basics

A matrix is a rectangular array of numbers. An m × n matrix has m rows and n columns.

Matrix Operations

Addition/Subtraction: Add or subtract corresponding elements (matrices must be the same size).

Scalar Multiplication: Multiply every element by the scalar.

Matrix Multiplication: For A (m × n) times B (n × p), the result C is (m × p): $C_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}$

Key Rule: Matrix multiplication is NOT commutative. AB ≠ BA in general. However, it IS associative: (AB)C = A(BC).

Transpose: Aᵀ swaps rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ

Special Matrices

Determinants

2 × 2 Determinant

$\det \begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

3 × 3 Determinant (Cofactor Expansion)

Expand along the first row: $\det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$

Properties of Determinants

• det(AB) = det(A) · det(B)
• det(Aᵀ) = det(A)
• det(kA) = kⁿ det(A) for n × n matrix
• Swapping two rows changes the sign of the determinant
• If two rows are identical, det = 0
• det(A) = 0 means A is singular (not invertible)

Solving Systems of Linear Equations

Matrix Form: Ax = b

For a system of n equations with n unknowns: $\begin{pmatrix} a_{11} & a_{12} & \cdots \\ a_{21} & a_{22} & \cdots \\ \vdots & & \ddots \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ \vdots \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ \vdots \end{pmatrix}$

Cramer's Rule

For Ax = b where det(A) ≠ 0: $x_i = \frac{\det(A_i)}{\det(A)}$ where Aᵢ is A with column i replaced by b.

Example: Solve 2x + 3y = 8, x - y = 1

det(A) = 2(-1) - 3(1) = -5

x = det([8, 3; 1, -1]) / (-5) = (-8 - 3)/(-5) = -11/(-5) = 11/5

y = det([2, 8; 1, 1]) / (-5) = (2 - 8)/(-5) = -6/(-5) = 6/5

Gaussian Elimination

Convert the augmented matrix [A|b] to row echelon form using elementary row operations, then back-substitute.

Inverse of a Matrix

For a 2 × 2 matrix: $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

The inverse exists only when det(A) ≠ 0.

Properties:

• A · A⁻¹ = A⁻¹ · A = I
• (AB)⁻¹ = B⁻¹A⁻¹ (reverse order!)
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ

Eigenvalues and Eigenvectors

For a square matrix A, eigenvalue λ and eigenvector v satisfy: $Av = \lambda v$

Finding eigenvalues: Solve det(A - λI) = 0 (the characteristic equation)

Example: For A = [4, 1; 2, 3]:

det(A - λI) = (4-λ)(3-λ) - (1)(2) = λ² - 7λ + 10 = (λ-5)(λ-2) = 0

Eigenvalues: λ₁ = 5, λ₂ = 2

Engineering Applications of Eigenvalues

• Structural analysis: Natural frequencies of vibration
• Control systems: System stability (all eigenvalues with negative real parts → stable)
• Principal stresses: Eigenvalues of the stress tensor