Matrices & Determinants — NDA Mathematics Formulas and Traps

Matrices: Order, Algebra, Powers & Equations

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

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

Common traps

Matrix multiplication is NOT commutative — $AB \neq BA$

Order matters:

AB

and

BA

are generally different matrices (and one may not even exist). Never reorder factors inside a product. If a question gives

AB

, compute

AB

— answering with

BA

is the classic trap.

$(AB)^T = B^T A^T$ — the order REVERSES

The transpose of a product flips the factors:

(AB)^T = B^T A^T

, not

A^T B^T

. Sum and scalar transposes keep their order (

(A+B)^T = A^T+B^T

); only the PRODUCT reverses. Writing

A^T B^T

is the trap.

Don't import $a^2 - b^2 = (a+b)(a-b)$ into matrices

Every number identity that secretly uses commutativity can break for matrices.

(A+B)(A-B)

,

(A+B)^2

,

(AB)^2 = A^2B^2

all FAIL unless

AB = BA

. When an option assumes one of these, it's almost always the trap answer.

$AB = O$ does NOT force $A = O$ or $B = O$

Unlike numbers, matrices have zero divisors: two non-zero matrices can multiply to the zero matrix (e.g.

\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} = O

). So you may NOT 'cancel' a matrix; concluding

A=O

or

B=O

from

AB=O

is the trap.

Special Matrices and Their Tell-Tale Properties

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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)}

The special-matrix catalog

Type	Defining property	Key consequence
Symmetric	$A^T = A$	$a_{ij} = a_{ji}$ ; inverse (if any) is symmetric
Skew-symmetric	$A^T = -A$	diagonal all 0; odd order $\Rightarrow \det = 0$ Odd-order skew-symmetric is ALWAYS singular (det 0). Even-order need not be.
Diagonal	off-diagonal all 0	$\det =$ product of diagonal entries
Orthogonal	$AA^T = I$	$A^{-1} = A^T$ ; $\det = \pm 1$
Idempotent	$A^2 = A$	$\det \in \{0, 1\}$
Involutory	$A^2 = I$	$A^{-1} = A$ ; $\det = \pm 1$
Hermitian	$(\bar{A})^T = A$	complex analogue of symmetric; diagonal entries are real; $A+(\bar{A})^T$ is always Hermitian For a REAL matrix, Hermitian $=$ symmetric. The conjugate-transpose $(\bar{A})^T$ is also written $A^{*}$ or $A^{\dagger}$ .
Skew-Hermitian	$(\bar{A})^T = -A$	complex analogue of skew-symmetric; diagonal entries are 0 or purely imaginary

Recognise the defining equation first; the determinant and inverse follow immediately.

Common traps

Odd-order skew-symmetric $\Rightarrow \det = 0$ ; diagonal entries are 0

From

A^T = -A

: the diagonal satisfies

a_{ii} = -a_{ii}

, so every diagonal entry is 0. And for ODD order,

\det A = (-1)^n\det A = -\det A

, forcing

\det A = 0

. Don't assume the determinant is unknown — for odd order it is always 0. (Even order need not be.)

The cross-product matrix of $v$ kills the outer product $vv^T$

The

3\times3

skew-symmetric matrix

P

with

Px = v\times x

(entries

0, -c, b;\ c, 0, -a;\ -b, a, 0

for

v=(a,b,c)

) and the symmetric matrix

Q = vv^T

satisfy

PQ = QP = O

: each column of

Q

is a multiple of

v

, and

v\times v = 0

. A statement-check asking whether

PQ

is zero, symmetric, or equal to

QP

has all three true.

Determinants: Evaluation & Properties

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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}

Common traps

$\det(kA) = k^n \det A$ , NOT $k\det A$

Pulling a scalar out of a determinant works one ROW at a time. Factoring

k

from the whole

n\times n

matrix factors it from each of the

n

rows →

k^n

. The single most common determinant mistake in this bank.

$\det(I+AA^T) = 1 + A^TA$ — don't forget the $+1$

For a column vector

A

,

\det(I+AA^T)=1+A^TA=1+\sum a_i^2

. The common slip is to report just

\sum a_i^2

(dropping the

+1

), or to expand the full

3\times3

by hand.

$\det(A+B) \neq \det A + \det B$

Determinant is multiplicative (

\det(AB)=\det A\,\det B

) but NOT additive.

\det(A+B)

has no shortcut — you must add the matrices first, then take ONE determinant. Splitting it as

\det A + \det B

is the trap.

It is a SUM of determinants, not one determinant with every row differentiated

\dfrac{d}{dx}\det\neq

the determinant of all-rows-differentiated. Differentiate exactly one row per term and add. For higher derivatives, apply the rule again to each surviving term.

Don't evaluate the coefficients — use Pascal to collapse a column

Computing

\binom{9}{4}

,

\binom{11}{6}

, … and expanding is slow and error-prone. The whole design is that one column is the Pascal sum of two others; the right move is a single column operation.

Special Determinants: Trig, Complex, ω, Polynomial

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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

Cofactors, Adjoint & Inverse

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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

\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

Common traps

$|\operatorname{adj}A| = |A|^{\,n-1}$ , NOT $|A|$ or $|A|^n$

The determinant of the adjoint carries the exponent

n-1

, where

n

is the ORDER. For a

3\times3

matrix it is

|A|^2

— students who answer

|A|

(forgetting the power) or

|A|^3

(over-counting) walk into the two standard distractors.

$\operatorname{adj}(AB) = \operatorname{adj}B\,\operatorname{adj}A$ — the order REVERSES

Like transpose and inverse, the adjoint of a product flips the factors:

\operatorname{adj}(AB) = \operatorname{adj}B\,\operatorname{adj}A

, not

\operatorname{adj}A\,\operatorname{adj}B

. Keeping the original order is the trap.

$(AB)^{-1} = B^{-1}A^{-1}$ , NOT $A^{-1}B^{-1}$

The inverse of a product reverses the order of factors — the single most-missed inverse fact. It must, so that

(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = I

. Writing

A^{-1}B^{-1}

is the trap.

$(A^T)^{-1} = (A^{-1})^T$ — don't drop the transpose

Transpose and inverse commute:

(A^T)^{-1} = (A^{-1})^T

. The slip is to compute

A^{-1}

and forget to transpose it (or vice-versa), reporting plain

A^{-1}

as the answer.

Linear Systems: Consistency & Cramer's Rule

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Cramer's rule

x = \frac{\Delta_x}{\Delta}, \quad y = \frac{\Delta_y}{\Delta}, \quad z = \frac{\Delta_z}{\Delta}

Matrices & Determinants formulas

Matrices: Order, Algebra, Powers & Equations

Matrix multiplication and conformability

Transpose and its rules

Matrix polynomials and equations

Matrix algebra — where numbers' rules break

Common traps

Special Matrices and Their Tell-Tale Properties

Symmetric and skew-symmetric matrices

Diagonal, scalar, and identity matrices

Orthogonal matrices

Rotation matrices

Idempotent and involutory matrices

The special-matrix catalog

Common traps

Determinants: Evaluation & Properties

Evaluating 2×2 and 3×3 determinants

Determinant of products, scalars, and powers

Core row and column properties

Singular matrices and determinant equations

Factor-theorem and Vandermonde determinants

Cyclic determinants

Sums and sequences of determinants

Differentiating a Determinant

Binomial-Coefficient Determinants (Pascal's Identity)

Common traps

Special Determinants: Trig, Complex, ω, Polynomial

Determinants with complex entries

Cube-root-of-unity determinants

Cofactors, Adjoint & Inverse

Minors, cofactors, and expansion

The adjoint (adjugate)

Adjoint power formulas

Inverse via the adjoint

Inverse properties and the reversal law

Common traps

Linear Systems: Consistency & Cramer's Rule

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