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Formula Sheets / Maths

Chapter 8

Matrices and Determinants

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Matrix basics Determinants Adjoint and inverse Systems of linear equations

Matrix basics

Transpose
$(A^{T})^{T} = A,$ $A^{T} = - A$
Symmetric / skew
$A^{T} = A$ symmetric $;$ $A^{T} = - A$ skew-symmetric $;$ $A = \frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})$
Trace
$tr (A + B) = tr A + tr B,$ $tr (A B) = tr (B A)$
Special matrices
$A^{2} = A$ idempotent $,$ $A^{k} = O$ nilpotent $,$ $A^{2} = I$ involutory $,$ $A A^{T} = I$ orthogonal

Determinants

Properties
$∣ A^{T} ∣ = ∣ A ∣,$ $∣ A B ∣ = ∣ A ∣∣ B ∣,$ $∣ k A ∣ = k^{n} ∣ A ∣ (A$ is $n \times n)$
Row operations
swap two rows: sign changes; two equal rows: $∣ A ∣ = 0; R_{i} \to R_{i} + k R_{j} :$ no change
Area of a triangle
$Δ = \frac{1}{2} x_{1} x_{2} x_{3} y_{1} y_{2} y_{3} 111$

Adjoint and inverse

Inverse
$A^{- 1} = \frac{adj A}{∣ A ∣},$ $∣ A ∣ \neq = 0$
Adjoint
$adj (adj A) = ∣ A ∣^{n - 2} A$ $∣ adj A ∣ = ∣ A ∣^{n - 1},$ $adj (adj A) = ∣ A ∣^{n - 2} A$
Rules
$(A B)^{- 1} = B^{- 1} A^{- 1},$ $∣ A^{- 1} ∣ = \frac{1}{∣ A ∣},$ $adj (A B) = adj B adj A$

Systems of linear equations

Cramer's rule
$x = \frac{Δ _{1}}{Δ}, y = \frac{Δ _{2}}{Δ}, z = \frac{Δ _{3}}{Δ}$
Consistency
$Δ \neq = 0 :$ unique solution $;$ $Δ = 0,$ some $Δ_{i} \neq = 0 :$ no solution $;$ $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0 :$ infinitely many or none
Homogeneous
$A X = O : Δ \neq = 0 \Rightarrow$ only $X = O;$ $Δ = 0 \Rightarrow$ non-trivial solutions

Binomial Theorem Trigonometric Ratios and Identities