Matrices — JEE Mains Mathematics

Formulas (6)

• Order, equality, linear combinations, and trace · Trace (sum of the diagonal)
  $$\operatorname{tr}(A)=\sum_{i=1}^{n} a_{ii}$$
• Matrix multiplication: conformability and the row-by-column rule · Row-by-column entry rule
  $$(AB)_{ij}=\sum_{k} a_{ik}\,b_{kj}$$
• Transpose and the reversal law · Transpose rules
  $$(AB)^{T}=B^{T}A^{T}\qquad (A^{T})^{T}=A$$
• Non-commutativity, zero divisors, and commuting matrices · Non-commutative square expansion
  $$(A+B)^{2}=A^{2}+AB+BA+B^{2}$$
• Elementary row operations · The three operations
  $$R_i\leftrightarrow R_j\qquad R_i\to kR_i\ (k\neq0)\qquad R_i\to R_i+kR_j$$
• Counting matrices · Counting bridges
  $$\#=k^{mn}\qquad \operatorname{tr}(A^{T}A)=\sum_{i,j}a_{ij}^{2}$$

Watch out for (5)

• Trace is the DIAGONAL sum, not the sum of all entries
  → Order, equality, linear combinations, and trace
• Matrix multiplication is NOT commutative — $AB\neq BA$
  → Matrix multiplication: conformability and the row-by-column rule
• $(AB)^{T}=B^{T}A^{T}$ — the order REVERSES
  → Transpose and the reversal law
• $AB=O$ does NOT force $A=O$ or $B=O$
  → Non-commutativity, zero divisors, and commuting matrices
• Don't import $a^{2}-b^{2}=(a+b)(a-b)$ into matrices
  → Non-commutativity, zero divisors, and commuting matrices

Formulas (9)

• Cyclic and pattern powers · Cyclic reduction
  $$A^d = I \ \Rightarrow\ A^n = A^{\,n \bmod d}$$
• Nilpotent shift: A = I + N · Binomial for a nilpotent shift
  $$A = I + N,\ N^k = O \ \Rightarrow\ A^n = I + nN + \binom{n}{2}N^2 + \dots$$
• Idempotent (A² = A) and involutory (A² = I) matrices · Idempotent binomial collapse
  $$A^2 = A \ \Rightarrow\ (I+A)^n = I + (2^n - 1)A$$
• The Cayley-Hamilton equation · Cayley-Hamilton (2×2)
  $$A^2 - (\operatorname{tr}A)\,A + (\det A)\,I = O$$
• Reducing higher powers with a polynomial relation · Power reduction
  $$A^2 = pA + qI \ \Rightarrow\ A^3 = (p^2+q)A + pq\,I$$
• Sums of powers of a matrix · Geometric sum for a rank-1 matrix
  $$A^2 = cA \ \Rightarrow\ \sum_{k=1}^{m}A^k = \frac{c^m - 1}{c - 1}\,A$$
• Powers under conjugation: (P⁻¹BP)ⁿ · Conjugation commutes with powers
  $$(P^{-1}BP)^n = P^{-1}B^{n}P \qquad (P^{T}P = I \Rightarrow P^{T}(PBP^{T})^{n}P = B^{n})$$
• Counting n with Aⁿ = A or Aⁿ = I · Counting the exponents
  $$A^n = I \iff d \mid n \ \Rightarrow\ \#\{n \le N\} = \left\lfloor \tfrac{N}{d} \right\rfloor$$
• Eigenvalue and trace-determinant reasoning · Sum and product of eigenvalues
  $$\lambda_1 + \lambda_2 = \operatorname{tr}A,\qquad \lambda_1\lambda_2 = \det A$$

Watch out for (11)

• Reduce the exponent modulo the period FIRST
  → Cyclic and pattern powers
• The binomial TRUNCATES — don't chase infinitely many terms
  → Nilpotent shift: A = I + N
• This is not a normal binomial — $I$ and $N$ must commute
  → Nilpotent shift: A = I + N
• $(I+A)^n = I + (2^n-1)A$ needs $A^2 = A$
  → Idempotent (A² = A) and involutory (A² = I) matrices
• The determinant term is $+\det A$ for a $2\times2$
  → The Cayley-Hamilton equation
• Cayley-Hamilton is about the CHARACTERISTIC polynomial
  → The Cayley-Hamilton equation
• Substitute the relation at every step — don't expand entrywise
  → Reducing higher powers with a polynomial relation
• Spot the rank-1 structure before you multiply
  → Sums of powers of a matrix
• Strip the conjugation FIRST
  → Powers under conjugation: (P⁻¹BP)ⁿ
• Find the PERIOD, then count residues — don't test each n
  → Counting n with Aⁿ = A or Aⁿ = I
• $\operatorname{tr}(A^2) \ne (\operatorname{tr}A)^2$
  → Eigenvalue and trace-determinant reasoning

Formulas (5)

• Definitions, entry patterns, and counting · Free-entry counts for order n
  $$\#\{\text{symmetric}\} = k^{\,n(n+1)/2}, \qquad \#\{\text{skew}\} = \begin{cases} k^{\,n(n-1)/2} & 0 \in \text{value set}\\[2pt] 0 & 0 \notin \text{value set}\end{cases}$$
• The symmetric-plus-skew decomposition · Unique symmetric + skew split
  $$A = \underbrace{\tfrac12(A+A^T)}_{P\ \text{symmetric}} + \underbrace{\tfrac12(A-A^T)}_{Q\ \text{skew}}$$
• Transposing products of symmetric and skew matrices · Transpose rules and skew-power parity
  $$(XY)^T = Y^T X^T, \qquad (B^n)^T = (-1)^n B^n \ \ (B\ \text{skew})$$
• The quadratic form test for skew-symmetry · Quadratic-form characterisation
  $$X^TAX = 0 \ \ \forall X \;\Longrightarrow\; A^T = -A \ \ (A\ \text{skew-symmetric})$$
• Orthogonal, rotation, and Cayley-transform matrices · Orthogonality and rotation composition
  $$AA^T=I \;\Rightarrow\; A^{-1}=A^T,\ \det A=\pm1 \qquad f(x)f(y)=f(x+y)$$

Watch out for (7)

• A symmetric matrix is NOT determined by $n^2$ free choices
  → Definitions, entry patterns, and counting
• Skew-symmetric over a set missing $0$ gives ZERO matrices
  → Definitions, entry patterns, and counting
• $B^n$ flips type with the parity of $n$
  → Transposing products of symmetric and skew matrices
• A product of symmetric matrices need not be symmetric
  → Transposing products of symmetric and skew matrices
• $X^TAX=0$ does not mean $A=O$
  → The quadratic form test for skew-symmetry
• $\det A=\pm1$, not always $+1$
  → Orthogonal, rotation, and Cayley-transform matrices
• Orthogonal is about $A^T=A^{-1}$, not $A^T=A$
  → Orthogonal, rotation, and Cayley-transform matrices

Formulas (7)

• Determinant of products, transposes and scalar multiples · Determinant identities
  $$\det(AB)=\det A\,\det B,\quad \det(A^T)=\det A,\quad \det(kA)=k^{\,n}\det A$$
• Adjoint identities: A·adj A = |A|I and |adj A| = |A|ⁿ⁻¹ · Adjoint identities
  $$A(\operatorname{adj}A) = |A|\,I_n, \qquad |\operatorname{adj}A| = |A|^{\,n-1}, \qquad \operatorname{adj}(kA) = k^{\,n-1}\operatorname{adj}A$$
• Adjoint of the adjoint · Double adjoint
  $$\operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A, \qquad |\operatorname{adj}(\operatorname{adj}A)| = |A|^{\,(n-1)^2}$$
• Inverse from the adjoint and PQ = kI · Inverse and PQ = kI
  $$A^{-1} = \frac{\operatorname{adj}A}{|A|}\ (|A|\neq0), \qquad PQ = kI_n \Rightarrow Q = kP^{-1},\ \ |Q| = \frac{k^{\,n}}{|P|}$$
• Inverse via Cayley–Hamilton (A⁻¹ = αA + βI) · 2×2 inverse from Cayley–Hamilton
  $$A^{2} - (\operatorname{tr}A)A + (\det A)I = O \ \Rightarrow\ A^{-1} = \frac{(\operatorname{tr}A)I - A}{\det A}$$
• When is a matrix invertible? (det ≠ 0) · Invertibility test
  $$A^{-1}\text{ exists} \iff |A| \neq 0 \qquad\text{and}\qquad AB = I \Rightarrow BA = I \ \text{(square matrices)}$$
• Special inverses, counting, and the reversal law · Reversal law and self-inverse
  $$(AB)^{-1} = B^{-1}A^{-1}, \qquad A = A^{-1} \iff A^{2} = I, \qquad \det(A + I) = \det A + \operatorname{tr}A + 1\ (2\times2)$$

Watch out for (10)

• $\det(kA) = k^{\,n}\det A$, NOT $k\det A$
  → Determinant of products, transposes and scalar multiples
• $|\operatorname{adj}A| = |A|^{\,n-1}$ — the exponent is $n-1$, NOT $n$
  → Adjoint identities: A·adj A = |A|I and |adj A| = |A|ⁿ⁻¹
• $\operatorname{adj}(kA) = k^{\,n-1}\operatorname{adj}A$ — note $n-1$, not $n$
  → Adjoint identities: A·adj A = |A|I and |adj A| = |A|ⁿ⁻¹
• $\operatorname{adj}(\operatorname{adj}A) = |A|^{\,n-2}A$: for $3\times3$ it's $|A|\cdot A$, for $2\times2$ it's just $A$
  → Adjoint of the adjoint
• $|\operatorname{adj}(\operatorname{adj}A)| = |A|^{4}$ for $3\times3$, NOT $|A|^2$
  → Adjoint of the adjoint
• $PQ = kI_n \Rightarrow |Q| = k^{\,n}/|P|$ — the $k^n$, not $k$
  → Inverse from the adjoint and PQ = kI
• Divide by $\det A$ — Cayley–Hamilton gives $A^{-1}$, not $\operatorname{adj}A$
  → Inverse via Cayley–Hamilton (A⁻¹ = αA + βI)
• $MN = O$ with $N$ invertible forces $M = O$ (so $|M| = 0$)
  → When is a matrix invertible? (det ≠ 0)
• $A = A^{-1}$ means $A^2 = I$, NOT $A = I$
  → Special inverses, counting, and the reversal law
• $\det(A + I) = \det A + \operatorname{tr}A + 1$ for $2\times2$, NOT $\det A + 1$
  → Special inverses, counting, and the reversal law