FE Mathematics: Linear Algebra Review

Linear algebra forms the computational backbone of nearly every engineering discipline, from structural analysis to control systems. For the FE exam, proficiency isn't about abstract theory—it's about executing fundamental matrix operations and solving systems of linear equations quickly and accurately. This review distills these concepts to their exam-essential forms, focusing on the problem types and sizes you are most likely to encounter.

Core Matrix Operations and Properties

At its heart, linear algebra deals with matrices, which are rectangular arrays of numbers. The most fundamental operations are addition and multiplication. Matrix addition is straightforward: you can only add matrices of identical dimensions by summing their corresponding entries. For example, if $A=\left[\begin{array}{cc}1& 3\\ -2& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}4& -1\\ 5& 2\end{array}\right]$, then their sum is $A+B=\left[\begin{array}{cc}5& 2\\ 3& 2\end{array}\right]$.

Matrix multiplication is more nuanced. You can multiply an $m\times n$ matrix by an $n\times p$ matrix, resulting in an $m\times p$ matrix. The operation is not commutative; $AB$ is generally not equal to $BA$. The entry in the $i^{th}$ row and $j^{th}$ column of the product $C=AB$ is computed as the dot product of the $i^{th}$ row of $A$ and the $j^{th}$ column of $B$: $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+...+a_{in}{b}_{nj}$. On the FE exam, you'll most often multiply small (2x2 or 3x3) matrices or a matrix by a vector.

Two critical unary operations are the transpose and the inverse. The transpose of a matrix $A$, denoted $A^T$, is formed by swapping its rows and columns. The inverse of a square matrix $A$, denoted $A^{-1}$, is the unique matrix such that $A{A}^{-1}=A^{-1}A=I$, where $I$ is the identity matrix. A matrix is invertible (or non-singular) only if its determinant is non-zero. For a 2x2 matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the inverse is given by a key formula: $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$.

Determinants, Cramer's Rule, and Systems of Equations

The determinant is a scalar value that provides crucial information about a square matrix. For a 2x2 matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, the determinant is $det(A)=ad-bc$. For a 3x3 matrix, use the rule of Sarrus or cofactor expansion. A determinant of zero means the matrix is singular (non-invertible) and that any associated system of linear equations is either dependent or inconsistent.

One direct application of determinants is Cramer's Rule, a method for solving a system of $n$ linear equations with $n$ unknowns, provided the system's coefficient matrix $A$ is non-singular. For a system $A\vec{x}=\vec{b}$, the solution for the $i^{th}$ variable $x_i$ is: $$x_i=\frac{det(A_i)}{det(A)}$$ where $A_i$ is the matrix formed by replacing the $i^{th}$ column of $A$ with the constant vector $\vec{b}$. While computationally heavy for large systems, Cramer's Rule is perfectly suited for the 2x2 and 3x3 systems common on the FE exam.

More broadly, you can solve systems using Gaussian elimination (row reduction) or, for express solutions to small systems, by directly using the matrix inverse: $\vec{x}=A^{-1}\vec{b}$. It's vital to recognize the three possible outcomes for any system: a unique solution (independent system, $det(A)\ne 0$), infinitely many solutions (dependent system), or no solution (inconsistent system).

Eigenvalue and Eigenvector Problems

An eigenvalue problem is central to dynamics, stability analysis, and principal component analysis. For a square matrix $A$, an eigenvector $\vec{v}$ is a non-zero vector that, when multiplied by $A$, yields a scalar multiple of itself. That scalar is the corresponding eigenvalue $\lambda$. This relationship is defined by the equation: $$A\vec{v}=\lambda \vec{v}$$ which can be rearranged to the standard form: $(A-\lambda I)\vec{v}=\vec{0}$.

For this homogeneous equation to have a non-trivial solution ($\vec{v}\ne \vec{0}$), the matrix $(A-\lambda I)$ must be singular. Therefore, finding eigenvalues requires solving the characteristic equation: $$det(A-\lambda I)=0$$ This results in a polynomial in $\lambda$. For a 2x2 matrix, it will be a quadratic; for the FE exam, expect to solve for these eigenvalues. Once an eigenvalue $\lambda$ is found, you find its corresponding eigenvector by solving $(A-\lambda I)\vec{v}=\vec{0}$ for the non-zero vector $\vec{v}$.

Common Pitfalls

1. Dimension Mismatch in Multiplication: The most frequent error is attempting to multiply matrices where the number of columns in the first does not equal the number of rows in the second. Always check dimensions first: $(m\times n)\cdot (n\times p)$ is valid and yields an $(m\times p)$ matrix.
2. Misapplying Cramer's Rule: Cramer's Rule only applies to systems with a square, non-singular coefficient matrix (i.e., $det(A)\ne 0$). Applying it to a system with a zero determinant or a non-square system is a critical mistake. Always check $det(A)$ first.
3. Confusing the Transpose and Inverse: The transpose and inverse are fundamentally different. The transpose simply flips rows and columns, while the inverse is a more complex operation that effectively "undoes" the matrix. Remember that $(AB{)}^T=B^T{A}^T$, but $(AB{)}^{-1}=B^{-1}{A}^{-1}$.
4. Algebraic Errors in the Characteristic Equation: When setting up $det(A-\lambda I)=0$, ensure you correctly subtract $\lambda$ from the diagonal entries only. A sign error here will derail the entire eigenvalue calculation. Double-check the subtraction before computing the determinant.

Summary

• Matrix operations are rule-specific: addition requires identical dimensions, multiplication requires the inner dimensions to match, and the inverse exists only for square matrices with a non-zero determinant.
• The determinant determines key properties; a value of zero indicates a singular matrix, meaning no unique inverse and no unique solution to $A\vec{x}=\vec{b}$.
• Cramer's Rule provides a determinant-based method to solve small systems ($n\le 3$) with a unique solution, but always verify $det(A)\ne 0$ first.
• Eigenvalues $\lambda$ are found by solving $det(A-\lambda I)=0$. For each eigenvalue, eigenvectors are the non-zero solutions to $(A-\lambda I)\vec{v}=\vec{0}$.
• On the FE exam, focus on fluent, error-free execution with 2x2 and 3x3 matrices, as these sizes represent the vast majority of exam questions.