Matrices & Determinants — NDA Mathematics

Formulas (5)

• Matrix multiplication and conformability · Order of a product
  $$A_{m\times n}\, B_{n\times p} = (AB)_{m\times p}$$
• Transpose and its rules · Transpose rules
  $$(AB)^T = B^T A^T$$
• Powers of a matrix · Powers of a matrix
  $$A^2 = A\cdot A \qquad A^n = A^{n-1}A$$
• Matrix polynomials and equations · Cayley–Hamilton (2×2)
  $$A^2 - (a+d)\,A + (ad - bc)\,I = O$$
• Matrix algebra — where numbers' rules break · Non-commutative expansions
  $$(A+B)^2 = A^2 + AB + BA + B^2 \qquad (A+B)(A-B) = A^2 - AB + BA - B^2$$

Watch out for (4)

• Matrix multiplication is NOT commutative — $AB \neq BA$
  → Matrix multiplication and conformability
• $(AB)^T = B^T A^T$ — the order REVERSES
  → Transpose and its rules
• Don't import $a^2 - b^2 = (a+b)(a-b)$ into matrices
  → Matrix algebra — where numbers' rules break
• $AB = O$ does NOT force $A = O$ or $B = O$
  → Matrix algebra — where numbers' rules break

Formulas (5)

• Symmetric and skew-symmetric matrices · Symmetric + skew decomposition
  $$A = \underbrace{\tfrac12(A + A^T)}_{\text{symmetric}} + \underbrace{\tfrac12(A - A^T)}_{\text{skew-symmetric}}$$
• Diagonal, scalar, and identity matrices · Diagonal determinant and inverse
  $$\det D = \prod_i d_i, \qquad D^{-1} = \operatorname{diag}\!\left(\tfrac{1}{d_1}, \dots, \tfrac{1}{d_n}\right) \qquad D^k = \operatorname{diag}(d_1^k, \dots, d_n^k) \qquad \det(kI_n) = k^n$$
• Orthogonal matrices · Orthogonality
  $$AA^T = I \;\Longrightarrow\; A^{-1} = A^T,\quad \det A = \pm 1 \qquad AA^T = A^T A = I$$
• Rotation matrices · Rotation composition
  $$R(\theta)\,R(\phi) = R(\theta + \phi), \qquad R(\theta)^n = R(n\theta)$$
• Idempotent and involutory matrices · Idempotent, involutory, all-ones, nilpotent
  $$A^2 = A \Rightarrow A^n = A \qquad A^2 = I \Rightarrow A^{-1} = A \qquad J_n^2 = nJ_n \qquad A^k = O \text{ (nilpotent)}$$

Reference tables (1)

Watch out for (2)

• Odd-order skew-symmetric $\Rightarrow \det = 0$; diagonal entries are 0
  → Symmetric and skew-symmetric matrices
• The cross-product matrix of $v$ kills the outer product $vv^T$
  → Symmetric and skew-symmetric matrices

Formulas (9)

• Evaluating 2×2 and 3×3 determinants · 2×2 determinant
  $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$
• Determinant of products, scalars, and powers · Multiplicativity and scaling
  $$\det(AB) = \det A\,\det B, \qquad \det(kA) = k^{\,n}\det A \qquad \det(A^T) = \det A \qquad \det(A^m) = (\det A)^m \qquad \det(A^{-1}) = \frac{1}{\det A} \qquad \det(B^{-1}AB) = \det A$$
• Core row and column properties · Core row/column properties
  $$R_i \leftrightarrow R_j \Rightarrow \det \to -\det \qquad \text{two identical/proportional rows} \Rightarrow \det = 0 \qquad \det(kR_i\text{-scaled}) = k\det A$$
• Singular matrices and determinant equations · Singular matrix
  $$A \text{ singular} \iff |A| = 0 \qquad |A| = 0 \Rightarrow A^{-1} \text{ does not exist}$$
• Factor-theorem and Vandermonde determinants · Vandermonde (3×3)
  $$\begin{vmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{vmatrix} = (a-b)(b-c)(c-a)$$
• Cyclic determinants · Cyclic determinant
  $$\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} = -(a^3+b^3+c^3-3abc)$$
• Sums and sequences of determinants · Determinant is not additive
  $$\det(A+B) \neq \det A + \det B \text{ (in general)}$$
• Differentiating a Determinant · Derivative of a 3-row determinant
  $$\frac{d}{dx}\begin{vmatrix}R_1\\R_2\\R_3\end{vmatrix} = \begin{vmatrix}R_1'\\R_2\\R_3\end{vmatrix} + \begin{vmatrix}R_1\\R_2'\\R_3\end{vmatrix} + \begin{vmatrix}R_1\\R_2\\R_3'\end{vmatrix}$$
• Binomial-Coefficient Determinants (Pascal's Identity) · Pascal's identity (the only tool needed)
  $$\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$$

Watch out for (5)

• $\det(kA) = k^n \det A$, NOT $k\det A$
  → Determinant of products, scalars, and powers
• $\det(I+AA^T) = 1 + A^TA$ — don't forget the $+1$
  → Determinant of products, scalars, and powers
• $\det(A+B) \neq \det A + \det B$
  → Determinant of products, scalars, and powers
• It is a SUM of determinants, not one determinant with every row differentiated
  → Differentiating a Determinant
• Don't evaluate the coefficients — use Pascal to collapse a column
  → Binomial-Coefficient Determinants (Pascal's Identity)

Formulas (2)

• Determinants with complex entries · Powers of i (period 4)
  $$i = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1$$
• Cube-root-of-unity determinants · Cube roots of unity
  $$\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0$$

Formulas (6)

• Minors, cofactors, and expansion · Cofactor expansion vs alien cofactors
  $$\sum_j a_{ij}C_{ij} = \det A, \qquad \sum_j a_{ij}C_{kj} = 0\ (k \neq i)$$
• The adjoint (adjugate) · Adjoint identity
  $$A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I_n$$
• Adjoint power formulas · Adjoint of an n×n matrix
  $$|\operatorname{adj}A| = |A|^{\,n-1}, \qquad \operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A$$
• Inverse via the adjoint · Inverse formula
  $$A^{-1} = \frac{1}{|A|}\operatorname{adj}A \quad (|A| \neq 0)$$
• Inverse properties and the reversal law · Reversal law for inverses
  $$(AB)^{-1} = B^{-1}A^{-1}$$
• Inverses of special matrices · Inverses of special matrices
  $$\operatorname{diag}(d_i)^{-1} = \operatorname{diag}(1/d_i) \qquad \text{orthogonal: } A^{-1} = A^T \qquad R(\theta)^{-1} = R(-\theta) \qquad \text{involutory: } A^{-1} = A$$

Watch out for (4)

• $|\operatorname{adj}A| = |A|^{\,n-1}$, NOT $|A|$ or $|A|^n$
  → Adjoint power formulas
• $\operatorname{adj}(AB) = \operatorname{adj}B\,\operatorname{adj}A$ — the order REVERSES
  → Adjoint power formulas
• $(AB)^{-1} = B^{-1}A^{-1}$, NOT $A^{-1}B^{-1}$
  → Inverse properties and the reversal law
• $(A^T)^{-1} = (A^{-1})^T$ — don't drop the transpose
  → Inverse properties and the reversal law

Formulas (1)

• Cramer's rule · Cramer's rule
  $$x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}$$