๐ Matrix Formulas Cheat Sheet
Matrix Formulas Cheat Sheet
Every essential linear algebra formula in one place. Bookmark this page or print it for your exams and problem sets.
๐ Determinants
2ร2 Determinant
$$\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$$
3ร3 Sarrus Rule
$$\det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$
Cofactor Expansion (Row i)
$$\det(A)=\sum_{j=1}^n a_{ij}(-1)^{i+j}M_{ij}$$
$M_{ij}$ = minor (det of matrix with row i, col j removed)
Determinant Properties
$$\det(AB)=\det(A)\det(B)$$
$$\det(A^T)=\det(A)$$
$$\det(kA)=k^n\det(A)$$
$$\det(A^{-1})=\frac{1}{\det(A)}$$
๐ Matrix Inverse
2ร2 Inverse
$$\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$$
General Inverse (Gauss-Jordan)
$$[A|I] \xrightarrow{\text{RREF}} [I|A^{-1}]$$
Aโปยน exists iff det(A) โ 0
Inverse Properties
$$AA^{-1}=A^{-1}A=I$$
$$(AB)^{-1}=B^{-1}A^{-1}$$
$$(A^T)^{-1}=(A^{-1})^T$$
$$(A^{-1})^{-1}=A$$
๐ Eigenvalues & Eigenvectors
Definition
$$A\mathbf{v}=\lambda\mathbf{v}, \quad \mathbf{v}\neq\mathbf{0}$$
Characteristic Equation
$$\det(A-\lambda I)=0$$
Roots ฮป are eigenvalues; solve (AโฮปI)v=0 for eigenvectors
Spectral Properties
$$\text{tr}(A)=\sum_i \lambda_i$$
$$\det(A)=\prod_i \lambda_i$$
$$\lambda(A^n)=\lambda^n$$
$$\lambda(A^{-1})=1/\lambda$$
๐ Matrix Decompositions
LU Decomposition
$$A=LU \quad (PA=LU \text{ with pivoting})$$
L: lower triangular (1s on diagonal), U: upper triangular
QR Decomposition
$$A=QR$$
Q: orthogonal ($Q^TQ=I$), R: upper triangular
SVD
$$A=U\Sigma V^T$$
U, V orthogonal; ฮฃ diagonal (singular values ฯโ โฅ ฯโ โฅ ... โฅ 0)
Gram-Schmidt
$$\mathbf{u}_k=\mathbf{v}_k-\sum_{j
$$\mathbf{e}_k=\frac{\mathbf{u}_k}{\|\mathbf{u}_k\|}$$
๐ Vector Operations
Dot Product
$$\mathbf{a}\cdot\mathbf{b}=\sum_i a_i b_i=|\mathbf{a}||\mathbf{b}|\cos\theta$$
Cross Product (3D)
$$\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$$
$$|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta$$
Vector Norms
$$\|\mathbf{v}\|_2=\sqrt{\sum_i v_i^2} \quad \text{(Euclidean)}$$
$$\|\mathbf{v}\|_1=\sum_i |v_i| \quad \text{(Manhattan)}$$
๐ Special Matrix Types
Symmetric
$$A^T=A$$
Real eigenvalues. Orthogonally diagonalizable.
Orthogonal
$$Q^TQ=I \Rightarrow Q^T=Q^{-1}$$
det(Q) = ยฑ1. Preserves lengths and angles.
Positive Definite
$$\mathbf{x}^TA\mathbf{x}>0 \;\forall\;\mathbf{x}\neq 0$$
All eigenvalues positive. All leading minors positive.
Diagonal
$$D=\text{diag}(d_1,\ldots,d_n)$$
D^n = diag(dโโฟ,...,dโโฟ). det(D) = โdแตข.
๐ Rank, Nullity & Linear Systems
Rank-Nullity Theorem
$$\text{rank}(A)+\text{nullity}(A)=n$$
n = number of columns of A
Solution Types for Ax = b
$$\text{Unique: rank}(A)=\text{rank}([A|b])=n$$
$$\text{Infinite: rank}(A)=\text{rank}([A|b])
$$\text{None: rank}(A)<\text{rank}([A|b])$$
Cramer's Rule (nรn)
$$x_i=\frac{\det(A_i)}{\det(A)}$$
$A_i$ = A with column i replaced by b. Only for square non-singular systems.