Matrix Formulas — Operations, Determinant & Inverse Reference A complete reference of matrix operations and formulas. Covers matrix arithmetic, the transpose, determinants (2×2 and 3×3), inverses, the identity matrix, eigenvalues, and how to solve linear systems with matrices.
Matrix Notation m × n Matrix
A = [ a 11 ⋯ a 1 n ⋮ ⋱ ⋮ a m 1 ⋯ a m n ] A = \begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix} A = a 11 ⋮ a m 1 ⋯ ⋱ ⋯ a 1 n ⋮ a mn Entry
a i j is the entry in row i , column j a_{ij} \text{ is the entry in row } i,\ \text{column } j a ij is the entry in row i , column j Square Matrix
n × n (same number of rows and columns) n \times n \text{ (same number of rows and columns)} n × n (same number of rows and columns) Zero Matrix
0 : all entries are 0 \mathbf{0}: \text{all entries are } 0 0 : all entries are 0 Identity Matrix
I n = [ 1 0 ⋯ 0 0 1 ⋯ 0 ⋮ ⋱ ⋮ 0 0 ⋯ 1 ] I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} I n = 1 0 ⋮ 0 0 1 0 ⋯ ⋯ ⋱ ⋯ 0 0 ⋮ 1
Basic Matrix Operations Addition
( A + B ) i j = a i j + b i j (A + B)_{ij} = a_{ij} + b_{ij} ( A + B ) ij = a ij + b ij Scalar Multiplication
( c A ) i j = c ⋅ a i j (c A)_{ij} = c \cdot a_{ij} ( c A ) ij = c ⋅ a ij Transpose
( A T ) i j = a j i (A^T)_{ij} = a_{ji} ( A T ) ij = a j i Transpose Properties
( A B ) T = B T A T , ( A T ) T = A (AB)^T = B^T A^T,\ (A^T)^T = A ( A B ) T = B T A T , ( A T ) T = A Matrix Multiplication
( A B ) i j = ∑ k = 1 n a i k b k j (AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj} ( A B ) ij = k = 1 ∑ n a ik b k j Identity Property
A I = I A = A AI = IA = A A I = I A = A Non-Commutativity
A B ≠ B A in general AB \ne BA \text{ in general} A B = B A in general 2×2 Matrix Determinant & Inverse 2×2 Determinant
det [ a b c d ] = a d − b c \det\!\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc det [ a c b d ] = a d − b c 2×2 Inverse
A − 1 = 1 a d − b c [ d − b − c a ] A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} A − 1 = a d − b c 1 [ d − c − b a ] 2×2 Invertibility
A − 1 exists ⟺ det ( A ) ≠ 0 A^{-1} \text{ exists} \iff \det(A) \ne 0 A − 1 exists ⟺ det ( A ) = 0 3×3 Matrix Determinant Cofactor Expansion (Row 1)
det ( A ) = a 11 M 11 − a 12 M 12 + a 13 M 13 \det(A) = a_{11} M_{11} - a_{12} M_{12} + a_{13} M_{13} det ( A ) = a 11 M 11 − a 12 M 12 + a 13 M 13 Minor M_{ij}
M i j = det ( matrix with row i and column j deleted ) M_{ij} = \det(\text{matrix with row } i \text{ and column } j \text{ deleted}) M ij = det ( matrix with row i and column j deleted ) Cofactor C_{ij}
C i j = ( − 1 ) i + j M i j C_{ij} = (-1)^{i+j} M_{ij} C ij = ( − 1 ) i + j M ij General Determinant
det ( A ) = ∑ j = 1 n a i j C i j (any row) \det(A) = \sum_{j=1}^{n} a_{ij} C_{ij} \text{ (any row)} det ( A ) = j = 1 ∑ n a ij C ij (any row) Determinant Properties Product
det ( A B ) = det ( A ) det ( B ) \det(AB) = \det(A) \det(B) det ( A B ) = det ( A ) det ( B ) Transpose
det ( A T ) = det ( A ) \det(A^T) = \det(A) det ( A T ) = det ( A ) Inverse
det ( A − 1 ) = 1 det ( A ) \det(A^{-1}) = \frac{1}{\det(A)} det ( A − 1 ) = det ( A ) 1 Scalar Multiple
det ( c A ) = c n det ( A ) (for n × n ) \det(cA) = c^n \det(A) \text{ (for } n \times n\text{)} det ( c A ) = c n det ( A ) (for n × n ) Row Swap
Swapping two rows negates det ( A ) \text{Swapping two rows negates } \det(A) Swapping two rows negates det ( A ) Matrix Inverse (General) Adjugate Formula
A − 1 = 1 det ( A ) adj ( A ) A^{-1} = \frac{1}{\det(A)}\,\operatorname{adj}(A) A − 1 = det ( A ) 1 adj ( A ) Adjugate
adj ( A ) = C T (transpose of cofactor matrix) \operatorname{adj}(A) = C^T \text{ (transpose of cofactor matrix)} adj ( A ) = C T (transpose of cofactor matrix) Inverse Property
A A − 1 = A − 1 A = I A A^{-1} = A^{-1} A = I A A − 1 = A − 1 A = I Inverse of Product
( A B ) − 1 = B − 1 A − 1 (AB)^{-1} = B^{-1} A^{-1} ( A B ) − 1 = B − 1 A − 1 Solving Linear Systems Matrix Equation
A x ⃗ = b ⃗ A \vec{x} = \vec{b} A x = b Solution (if A is invertible)
x ⃗ = A − 1 b ⃗ \vec{x} = A^{-1} \vec{b} x = A − 1 b Cramer's Rule (x_i)
x i = det ( A i ) det ( A ) x_i = \frac{\det(A_i)}{\det(A)} x i = det ( A ) det ( A i ) Gaussian Elimination
[ A ∣ b ⃗ ] → row reduce [ I ∣ x ⃗ ] [A \mid \vec{b}] \xrightarrow{\text{row reduce}} [I \mid \vec{x}] [ A ∣ b ] row reduce [ I ∣ x ] Eigenvalues & Eigenvectors Eigenvalue Equation
A v ⃗ = λ v ⃗ A \vec{v} = \lambda \vec{v} A v = λ v Characteristic Equation
det ( A − λ I ) = 0 \det(A - \lambda I) = 0 det ( A − λ I ) = 0 Trace = Sum of Eigenvalues
tr ( A ) = ∑ λ i \operatorname{tr}(A) = \sum \lambda_i tr ( A ) = ∑ λ i Determinant = Product of Eigenvalues
det ( A ) = ∏ λ i \det(A) = \prod \lambda_i det ( A ) = ∏ λ i
