Formula index

Questions by the identity their solution uses

A different way into the same bank. Instead of filtering by chapter, pick the formula you want to drill and get every question whose solution actually invokes it — including the ones whose stem never mentions it.

Matrices & Determinants

40 identities across 707 questions, read and tagged one by one. Chapter notes.

Cofactor expansionTechnique $\det A = \sum_{j} a_{ij}C_{ij} \quad (\text{any fixed row } i)$ 99questions
Row and column operationsTechnique $R_i \to R_i + \lambda R_j \;\Rightarrow\; \det A \text{ unchanged}$ 71questions
Determinant of a 2×2 matrixFormula $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$ 65questions
Powers of a matrixTechnique $A^n = \underbrace{A\cdot A\cdots A}_{n}$ 54questions
Matrix polynomial equationsTechnique $A^2 - (\operatorname{tr}A)\,A + |A|\,I = O \quad (2\times2)$ 48questions
Determinants with trigonometric entriesTechnique $\sin^2\theta + \cos^2\theta = 1$ 47questions
Proportional rows make a determinant vanishProperty $R_i = \lambda R_j \;\Rightarrow\; \det A = 0$ 45questions
Determinant of the adjointFormula $|\operatorname{adj}A| = |A|^{\,n-1}$ 43questions
Inverse via the adjointFormula $A^{-1} = \frac{1}{|A|}\operatorname{adj}A \qquad (|A| \neq 0)$ 43questions
Skew-symmetric matricesProperty $A^T = -A \;\Rightarrow\; a_{ii} = 0$ 37questions
Symmetric matricesProperty $A^T = A$ 36questions
Determinant of a scalar multipleFormula $|kA| = k^{\,n}\,|A|$ 35questions
The adjoint identityFormula $A(\operatorname{adj}A) = (\operatorname{adj}A)A = |A|\,I_n$ 35questions
Order of a matrix productFormula $A_{m\times n}\,B_{n\times p} = (AB)_{m\times p}$ 34questions
Reversal law for transposesFormula $(AB)^T = B^T A^T$ 33questions
Definition of the inverseProperty $AB = BA = I \iff B = A^{-1}$ 33questions
Definition of the adjointProperty $(\operatorname{adj}A)_{ij} = C_{ji}$ 29questions
Consistency of a linear systemProperty $\Delta \neq 0 \Rightarrow \text{unique solution}$ 28questions
Singular matricesProperty $A \text{ singular} \iff |A| = 0 \iff A^{-1} \text{ does not exist}$ 28questions
Non-trivial solutions of a homogeneous systemProperty $AX = O \text{ has } X \neq O \iff \Delta = 0$ 24questions
Rules of transpositionFormula $(A^T)^T = A, \qquad (A+B)^T = A^T + B^T, \qquad (kA)^T = kA^T$ 23questions
Determinant of the inverseFormula $|A^{-1}| = \frac{1}{|A|}$ 21questions
Cube roots of unity in determinantsFormula $\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0$ 20questions
Determinant of a productFormula $|AB| = |A|\,|B|$ 18questions
Determinant of a powerFormula $|A^m| = |A|^{\,m}$ 18questions
Determinant equations and their rootsTechnique $\det A(x) = 0$ 17questions
Orthogonal matricesProperty $AA^T = I \iff A^{-1} = A^T$ 17questions
Idempotent matricesProperty $A^2 = A \;\Rightarrow\; A^k = A \ (k \ge 1)$ 17questions
Involutory matricesProperty $A^2 = I \iff A^{-1} = A$ 16questions
Equality of matricesProperty $A = B \iff a_{ij} = b_{ij} \ \forall\, i,j$ 16questions
Determinant of the transposeFormula $|A^T| = |A|$ 15questions
Solving AX = BTechnique $AX = B \;\Rightarrow\; X = A^{-1}B \quad (|A| \neq 0)$ 15questions
Matrix multiplication is not commutativeProperty $(A+B)(A-B) = A^2 - AB + BA - B^2$ 15questions
Diagonal matricesProperty $a_{ij} = 0 \ (i \neq j)$ 14questions
Adjoint of an adjointFormula $\operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}\,A$ 14questions
Determinant of a triangular matrixFormula $\det A = \prod_i a_{ii}$ 14questions
Scalar matricesProperty $A = kI$ 13questions
Addition and scalar multiplicationProperty $(A + B)_{ij} = a_{ij} + b_{ij}, \qquad (kA)_{ij} = k\,a_{ij}$ 13questions
Counting matricesTechnique $\text{orders of } N \text{ entries} = d(N), \qquad \text{fillings} = c^{\,mn}$ 12questions
Taking a common factor out of a rowProperty $\det(\ldots, kR_i, \ldots) = k\,\det(\ldots, R_i, \ldots)$ 12questions

Identities used by fewer than 12 questions in a chapter are not listed: a page with four questions on it is not a drill. This chapter yielded 79 distinct identities in all, of which 40 clear that bar.