Matrices formulas

Every formula JEE Advanced expects you to know for Matrices, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 34
Subject: Mathematics
Exam: JEE Advanced
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Matrices, every formula

34 formulas, typeset and free. Print it, or keep it open beside your practice.

Order of a matrix
$A_{m \times n} has mn elements$
m rows, n columns
Diagonal matrix power
$A = diag (d_{1}, \dots, d_{n}) \Rightarrow A^{k} = diag (d_{1}^{k}, \dots, d_{n}^{k})$
Trace
$tr (A) = i = 1 \sum n a_{ii}$
sum of diagonal elements
Trace linearity
$tr (A + B) = tr A + tr B, tr (k A) = k tr A$
Trace of product
$tr (A B) = tr (B A)$
Matrix product entry
$(A B)_{ij} = k = 1 \sum p a_{ik} b_{k j}$
A is m×p, B is p×n
Conformability
$A_{m \times p} B_{p \times n} = (A B)_{m \times n}$
columns of A = rows of B
Non-commutativity
$A B \neq = B A (in general)$
(A+B)^2
$(A + B)^{2} = A^{2} + B^{2} + A B + B A$
Transpose of product
$(A B)^{T} = B^{T} A^{T}$
Transpose of sum
$(A + B)^{T} = A^{T} + B^{T}, (k A)^{T} = k A^{T}$
Symmetric matrix
$A^{T} = A \Leftrightarrow a_{ij} = a_{j i}$
Skew-symmetric matrix
$A^{T} = - A \Rightarrow a_{ii} = 0, tr A = 0$
Symmetric + skew decomposition
$A = \frac{1}{2} (A + A^{T}) + \frac{1}{2} (A - A^{T})$
unique sym + skew split
Orthogonal matrix
$A A^{T} = I \Leftrightarrow A^{- 1} = A^{T}$
Idempotent matrix
$A^{2} = A$
Involutory matrix
$A^{2} = I$
Nilpotent matrix
$A^{p} = O for some p \in N$
Determinant 2×2
$det [a c b d] = a d - b c$
Cofactor
$A_{ij} = (- 1)^{i + j} M_{ij}$
M_{ij} = minor
Expansion along a row
$det A = a_{i 1} A_{i 1} + a_{i 2} A_{i 2} + a_{i 3} A_{i 3}$
Determinant of scalar multiple
$∣ k A ∣ = k^{n} ∣ A ∣$
n = order of A
Determinant of product
$∣ A B ∣ = ∣ A ∣ ∣ B ∣$
Determinant of transpose
$∣ A^{T} ∣ = ∣ A ∣$
A·adj A
$A (adj A) = ∣ A ∣ I_{n} = (adj A) A$
Determinant of adjoint
$∣ adj A ∣ = ∣ A ∣^{n - 1}$
adj of adj
$adj (adj A) = ∣ A ∣^{n - 2} A$
|A|≠0
Inverse via adjoint
$A^{- 1} = \frac{adj A}{∣ A ∣}$
|A|≠0
Inverse of product
$(A B)^{- 1} = B^{- 1} A^{- 1}$
Determinant of inverse
$∣ A^{- 1} ∣ = \frac{1}{∣ A ∣}$
Characteristic equation (3×3)
$λ^{3} - S_{1} λ^{2} + S_{2} λ - S_{3} = 0$
S1=tr, S3=|A|
Cramer's rule
$x = \frac{Δ _{1}}{Δ}, y = \frac{Δ _{2}}{Δ}, z = \frac{Δ _{3}}{Δ}$
Δ≠0
Homogeneous non-trivial solution
$A X = O has non-trivial solution \Leftrightarrow ∣ A ∣ = 0$
Vandermonde determinant
$111 a b c a^{2} b^{2} c^{2} = (a - b) (b - c) (c - a)$

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Matrices formulas

Matrices, every formula

Order of a matrix

Diagonal matrix power

Trace

Trace linearity

Trace of product

Matrix product entry

Conformability

Non-commutativity

(A+B)^2

Transpose of product

Transpose of sum

Symmetric matrix

Skew-symmetric matrix

Symmetric + skew decomposition

Orthogonal matrix

Idempotent matrix

Involutory matrix

Nilpotent matrix

Determinant 2×2

Cofactor

Expansion along a row

Determinant of scalar multiple

Determinant of product

Determinant of transpose

A·adj A

Determinant of adjoint

adj of adj

Inverse via adjoint

Inverse of product

Determinant of inverse

Characteristic equation (3×3)

Cramer's rule

Homogeneous non-trivial solution

Vandermonde determinant

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