Linear Algebra Fundamentals Cheat Sheet

Linear algebra fundamentals introduce students to matrices, vectors, systems of equations, and transformations. This cheat sheet helps organize the notation and rules that often appear in algebra, precalculus, computer science, and physics. Students need it because small errors in matrix size, order, or arithmetic can change an entire answer.

It is designed as a quick reference for checking procedures and formulas while solving problems.

The most important ideas are matrix dimensions, vector operations, matrix multiplication, determinants, inverses, and solving linear systems. A matrix with size $m \times n$ has $m$ rows and $n$ columns, and many operations only work when dimensions match. Matrix multiplication uses row by column dot products, so usually $AB \ne BA$ .

Determinants and inverses help decide whether a square matrix can be used to solve systems like $A\vec{x}=\vec{b}$ .

Key Facts

A matrix with $m$ rows and $n$ columns has size $m \times n$ , and its entry in row $i$ and column $j$ is written $a_{ij}$ .
Matrices can be added or subtracted only when they have the same dimensions, using $(A+B)_{ij}=a_{ij}+b_{ij}$ .
Scalar multiplication means multiplying every entry by the same number, so $(kA)_{ij}=ka_{ij}$ .
Matrix multiplication $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$ .
The entry in row $i$ and column $j$ of $AB$ is $(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}$ .
For vectors, the dot product is $\vec{u}\cdot\vec{v}=u_1v_1+u_2v_2+\cdots+u_nv_n$ .
For a $2\times2$ matrix $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ , the determinant is $\det(A)=ad-bc$ .
If $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ and $ad-bc\ne0$ , then $A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ .

Vocabulary

Matrix: A matrix is a rectangular array of numbers arranged in rows and columns.
Dimension: The dimension of a matrix is its size, written as rows by columns, such as $3\times2$ .
Vector: A vector is an ordered list of numbers that can represent a direction, a point, or a quantity with components.
Dot Product: The dot product combines two equal-length vectors by multiplying matching components and adding the results.
Determinant: The determinant is a number calculated from a square matrix that helps show whether the matrix has an inverse.
Inverse Matrix: An inverse matrix $A^{-1}$ reverses the effect of $A$ and satisfies $AA^{-1}=I$ when it exists.

Common Mistakes to Avoid

Adding matrices with different dimensions is wrong because matrix addition requires matching entries in the same positions.
Multiplying matrices in the wrong order is wrong because matrix multiplication is not usually commutative, so $AB$ and $BA$ may be different or one may be undefined.
Forgetting the row by column rule is wrong because each entry of $AB$ must come from a dot product of a row of $A$ with a column of $B$ .
Using the inverse formula when $\det(A)=0$ is wrong because a square matrix has no inverse if its determinant is zero.
Confusing rows and columns in matrix size is wrong because a $2\times3$ matrix has $2$ rows and $3$ columns, not the other way around.

Practice Questions

1 Find the size of $A=\begin{bmatrix}4&-1&2\\0&3&5\end{bmatrix}$ and identify the entry $a_{23}$ .
2 Compute $AB$ for $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2&0\\1&5\end{bmatrix}$ .
3 Find $\det(A)$ and decide whether $A^{-1}$ exists for $A=\begin{bmatrix}6&2\\3&1\end{bmatrix}$ .
4 Explain why the product of a $3\times2$ matrix and a $3\times3$ matrix is undefined, but the product of a $3\times3$ matrix and a $3\times2$ matrix is defined.

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Linear Algebra Fundamentals Cheat Sheet

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