Mathematics reference
Rules for matrices
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MathRef
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.
A0 == I equation 13
A2 == A A equation 14
An = A An - 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.
IT = I equation 19
(AT)T = A equation 20
(A + B)T = AT + BT equation 21
(r A)T = r AT equation 22
(A B)T = BT AT 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 = (AT)-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 AT 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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