Linear Algebra Matrices and Determinants Cheat Sheet

Key Facts

• Matrix addition and scalar multiplication are entrywise: if $A = [a_{ij}]$ and $B = [b_{ij}]$, then $A+B = [a_{ij}+b_{ij}]$ and $cA = [ca_{ij}]$.
• Matrix multiplication is defined by $(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$ when $A$ is $m \times n$ and $B$ is $n \times p$.
• The inverse of a nonsingular square matrix satisfies $AA^{-1} = A^{-1}A = I$, and for $2 \times 2$ matrices, $\begin{bmatrix}a&b\\c&d\end{bmatrix}^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$ when $ad-bc \ne 0$.
• For a $2 \times 2$ matrix, $\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$.
• A square matrix $A$ is invertible exactly when $\det(A) \ne 0$, $\operatorname{rank}(A)=n$, and the equation $Ax=0$ has only the trivial solution.
• The rank-nullity theorem states that for an $m \times n$ matrix $A$, $\operatorname{rank}(A)+\operatorname{nullity}(A)=n$.
• Eigenvalues satisfy the characteristic equation $\det(A-\lambda I)=0$, and eigenvectors satisfy $Av=\lambda v$ with $v \ne 0$.
• Common factorizations include $A=LU$ for elimination, $A=QR$ for orthogonal decomposition, and $A=U\Sigma V^{T}$ for singular value decomposition.

Vocabulary

Matrix

A rectangular array of numbers or symbols arranged in rows and columns, often used to represent a linear transformation or system of equations.

Determinant

A scalar value $\det(A)$ assigned to a square matrix that measures signed volume scaling and indicates whether the matrix is invertible.

Rank

The rank of a matrix is the dimension of its column space, equal to the number of pivot columns in its row echelon form.

Nullity

The nullity of a matrix is the dimension of the solution space of $Ax=0$, equal to the number of free variables.

Eigenvalue

An eigenvalue $\lambda$ is a scalar for which there exists a nonzero vector $v$ satisfying $Av=\lambda v$.

Factorization

A matrix factorization rewrites a matrix as a product of simpler matrices, such as $A=LU$, $A=QR$, or $A=U\Sigma V^{T}$.