Mathematics reference
Rules for matrices


Basic properties of matrices.


Legend.

 A, B, and C are matrices,
 O represents the zero matrix,
 I represents the identity matrix,
 r, s, and n are scalars.

Basic.


A == (1) A

equation 1


A  B == A + (B)

equation 2


1 A = A

equation 3


0 A = O

equation 4


A + O = O + A = A

equation 5


I A = A I = A

equation 6


A  A = O

equation 7


Addition and scalar product.


A + B = B + A

equation 8


(A + B) + C = A + (B + C)

equation 9


r (A + B) = r A + r B

equation 10


(r + s) A = r A + s A

equation 11


(r s) A = r (s A)

equation 12


Matrix product.


A^{0} == I

equation 13


A^{2} == A A

equation 14


A^{n} = A A^{n  1}

equation 15


(A B) C = A (B C)

equation 16


A (B + C) = A B + A C

equation 17


(A + B) C = A C + B C

equation 18


Transpose and inverse.


I^{T} = I

equation 19


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

equation 20


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

equation 21


(r A)^{T} = r A^{T}

equation 22


(A B)^{T} = B^{T} A^{T}

equation 23


I^{1} = I

equation 24


A A^{1} = A^{1} A = I

equation 25


(A B)^{1} = B^{1} A^{1}

equation 26


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

equation 27


Trace.


tr (A + B) = tr A + tr B

equation 28


tr (r A) = r tr A

equation 29


tr (A B) = tr (B A)

equation 30


Determinant and adjoint.


det O = 0

equation 31


det I = 1

equation 32


det A = det A^{T}

equation 33


det (A B) = (det A) (det B)

equation 34


A (adj A) = (adj A) A = (det A) I

equation 35


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