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Linear algebra formula sheet: determinants, inverses, rank and eigenvalues

The formulas that come back in a first linear algebra course, in one place and ready to print.

Determinants

Expression	Equals
$det (a c b d)$	$a d - b c$
$det (A B)$	$det A \cdot det B$
$det (A^{T})$	$det A$
$det (A^{- 1})$	$\frac{1}{det A}$
$det (c A), A \in R^{n \times n}$	$c^{n} det A$
det of a triangular matrix	$a_{11} a_{22} \dots a_{nn}$

Row operations and the determinant

Expression	Equals
$R_{i} \leftrightarrow R_{j}$	$det \to - det$
$R_{i} \to c R_{i}$	$det \to c det$
$R_{i} \to R_{i} + c R_{j}$	det unchanged

The inverse matrix

Expression	Equals
$(a c b d)^{- 1}$	$\frac{1}{a d - b c} (d - c - b a)$
$(A B)^{- 1}$	$B^{- 1} A^{- 1}$
$(A^{T})^{- 1}$	$(A^{- 1})^{T}$
$A^{- 1}$	$\frac{1}{det A} adj A$
A is invertible	$det A \neq = 0 ⟺ rank A = n$

Rank and solving systems

Expression	Equals
$rank A + dim ker A$	n, the number of columns
Ax = b has a solution	$rank A = rank [A ∣ b]$
The solution is unique	$rank A = rank [A ∣ b] = n$

Eigenvalues and diagonalisation

Expression	Equals
Eigenvalues	$det (A - λ I) = 0$
$λ_{1} + \dots + λ_{n}$	$tr A$
$λ_{1} \dots λ_{n}$	$det A$
$A = P D P^{- 1}$	$D = diag (λ_{i}), P = [v_{1} \dots v_{n}]$
$A^{k}$	$P D^{k} P^{- 1}$

Vectors, products and projection

Expression	Equals
$u \cdot v$	$\sum u_{i} v_{i} = ∣ u ∣∣ v ∣ cos θ$
$∣ u ∣$	$u \cdot u$
$proj_{v} u$	$\frac{u \cdot v}{v \cdot v} v$
$∣ u \times v ∣$	$∣ u ∣∣ v ∣ sin θ$
$∣ u \cdot v ∣$	$\leq ∣ u ∣∣ v ∣$

Notes

A quick check for eigenvalues: they add up to the trace and multiply to the determinant.

Diagonalising A = PDP⁻¹ works when there are n independent eigenvectors. Then powers of A are just powers of the diagonal.