1.4 Linear Algebra, Matrices, and Systems of Equations

1.4 Linear Algebra, Matrices, and Systems of Equations

Linear algebra provides the mathematical framework for solving large-scale structural truss systems, electrical circuit networks, finite element analysis (FEA), state-space control systems, and principal stress transformations on the FE exam.

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1. Matrix Operations & Properties

Matrix Definition & Order

An $m \times n$ matrix $A$ consists of $m$ rows and $n$ columns. The element in row $i$ and column $j$ is denoted $a_{ij}$.

Fundamental Operations

• Scalar Multiplication: $k A = [k a_{ij}]$

• Matrix Addition: For $A$ and $B$ of identical size $m \times n$, $C = A + B \implies c_{ij} = a_{ij} + b_{ij}$.

• Matrix Multiplication: For $A$ ($m \times p$) and $B$ ($p \times n$), the product $C = AB$ ($m \times n$) has elements:

  $$c_{ij} = \sum_{k=1}^{p} a_{ik} b_{kj}$$

  Note: Matrix multiplication is non-commutative ($AB \neq BA$ in general).

• Transpose of a Matrix ($A^T$): Formed by swapping rows and columns ($a_{ij}^T = a_{ji}$).

  • $(A + B)^T = A^T + B^T$
  • $(AB)^T = B^T A^T$
  • A matrix is symmetric if $A^T = A$, and skew-symmetric if $A^T = -A$.

• Trace of a Square Matrix: Sum of diagonal elements $\text{tr}(A) = \sum_{i=1}^n a_{ii}$.

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2. Determinants & Matrix Inverses

Determinants

Determinants exist only for square ($n \times n$) matrices.

• $2 \times 2$ Matrix:

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

• $3 \times 3$ Matrix (Cofactor Expansion along Row 1):

  $$\det \begin{bmatrix} a_{11} & a_{12} & a_{13} \\\ a_{21} & a_{22} & a_{23} \\\ a_{31} & a_{32} & a_{33} \end{bmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})$$

Key Determinant Rules

1. $\det(AB) = \det(A) \cdot \det(B)$
2. $\det(A^T) = \det(A)$
3. $\det(A^{-1}) = \frac{1}{\det(A)}$
4. $\det(kA) = k^n \det(A)$ for an $n \times n$ matrix.
5. Swapping two rows/columns changes the sign of the determinant.
6. If a row or column is all zeros, or if two rows are identical/proportional, $\det(A) = 0$.

Inverse Matrix ($A^{-1}$)

A square matrix $A$ has an inverse $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$ if and only if $\det(A) \neq 0$ ($A$ is non-singular).

• $2 \times 2$ Matrix Inverse Formula:

  $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\\ -c & a \end{bmatrix}$$

• General Adjugate Formula:

  $$A^{-1} = \frac{1}{\det(A)} \text{adj}(A) = \frac{1}{\det(A)} [C_{ij}]^T$$

  where $C_{ij} = (-1)^{i+j} M_{ij}$ is the cofactor matrix.

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3. Systems of Linear Equations & Cramer's Rule

A system of $n$ linear equations in $n$ unknowns can be written in matrix form $A\mathbf{x} = \mathbf{b}$:

$$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\\ a_{21} & a_{22} & \cdots & a_{2n} \\\ \vdots & \vdots & \ddots & \vdots \\\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix} \begin{bmatrix} x_1 \\\ x_2 \\\ \vdots \\\ x_n \end{bmatrix} = \begin{bmatrix} b_1 \\\ b_2 \\\ \vdots \\\ b_n \end{bmatrix}$$

Existence & Uniqueness Criteria

• Unique Solution: $\text{rank}(A) = \text{rank}([A|\mathbf{b}]) = n \iff \det(A) \neq 0$.
• Infinitely Many Solutions: $\text{rank}(A) = \text{rank}([A|\mathbf{b}]) < n$.
• No Solution (Inconsistent System): $\text{rank}(A) < \text{rank}([A|\mathbf{b}])$.

Cramer's Rule

If $\det(A) \neq 0$, the unique solution for variable $x_i$ is:

$$x_i = \frac{\det(A_i)}{\det(A)}$$

where $A_i$ is the matrix formed by replacing column $i$ of $A$ with column vector $\mathbf{b}$.

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4. Eigenvalues and Eigenvectors

For a square $n \times n$ matrix $A$, a scalar $\lambda$ and non-zero vector $\mathbf{v}$ are an eigenvalue and eigenvector if:

$$A \mathbf{v} = \lambda \mathbf{v} \implies (A - \lambda I)\mathbf{v} = \mathbf{0}$$

Characteristic Equation

For a non-trivial vector $\mathbf{v} \neq \mathbf{0}$ to exist, the matrix $(A - \lambda I)$ must be singular:

$$\det(A - \lambda I) = 0$$

Evaluating this determinant yields the $n$-th degree characteristic polynomial in $\lambda$. Roots of this polynomial are the eigenvalues $\lambda_i$.

Physical Significance in Engineering

• Solid Mechanics: Eigenvalues of the 2D/3D stress tensor represent principal stresses ($\sigma_1, \sigma_2, \sigma_3$), and eigenvectors indicate principal stress orientations.
• Vibrations & Dynamics: Eigenvalues represent squared natural frequencies ($\omega^2$), and eigenvectors represent mode shapes.

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5. Worked Engineering Problems

Worked Example 1: Solving Electrical Mesh System via Cramer's Rule

Problem: Solve for mesh current $I_2$ in the 3-mesh circuit system:

$$\begin{bmatrix} 5 & -2 & 0 \\\ -2 & 8 & -3 \\\ 0 & -3 & 6 \end{bmatrix} \begin{bmatrix} I_1 \\\ I_2 \\\ I_3 \end{bmatrix} = \begin{bmatrix} 12 \\\ 0 \\\ 0 \end{bmatrix}$$

Solution:

Step 1: Compute determinant of coefficient matrix $\det(A)$.

$$\det(A) = 5[(8)(6) - (-3)(-3)] - (-2)[(-2)(6) - (-3)(0)] + 0$$ $$\det(A) = 5[48 - 9] + 2[-12 - 0] = 5(39) - 24 = 195 - 24 = 171$$

Step 2: Form matrix $A_2$ by replacing Column 2 with vector $\mathbf{b}$.

$$A_2 = \begin{bmatrix} 5 & 12 & 0 \\\ -2 & 0 & -3 \\\ 0 & 0 & 6 \end{bmatrix}$$

Step 3: Compute $\det(A_2)$ by expanding along Row 3 or Column 2. Expanding along Column 2:

$$\det(A_2) = -12 \begin{vmatrix} -2 & -3 \\\ 0 & 6 \end{vmatrix} = -12 [(-2)(6) - (0)] = -12(-12) = 144$$

Step 4: Compute $I_2$.

$$I_2 = \frac{\det(A_2)}{\det(A)} = \frac{144}{171} = \frac{16}{19} \approx 0.8421\text{ A}$$

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Worked Example 2: Principal Stresses via Eigenvalue Analysis

Problem: A state of plane stress is represented by matrix $A = \begin{bmatrix} 40 & 20 \\\ 20 & 10 \end{bmatrix}\text{ MPa}$. Find the eigenvalues $\lambda_1, \lambda_2$ (principal stresses) of matrix $A$.

Solution:

Step 1: Set up characteristic equation $\det(A - \lambda I) = 0$.

$$\det \begin{bmatrix} 40 - \lambda & 20 \\\ 20 & 10 - \lambda \end{bmatrix} = (40 - \lambda)(10 - \lambda) - (20)(20) = 0$$ $$\lambda^2 - 50\lambda + 400 - 400 = 0$$ $$\lambda^2 - 50\lambda = 0$$

Step 2: Solve for roots $\lambda$.

$$\lambda(\lambda - 50) = 0$$ $$\lambda_1 = 50\text{ MPa}, \quad \lambda_2 = 0\text{ MPa}$$

Thus, the principal stresses are $\sigma_1 = 50\text{ MPa}$ and $\sigma_2 = 0\text{ MPa}$.