Properties of Matrix Operations
The main properties of matrix operations such as addition, multiplication, transpose and inverse are presented.
In what follows, A, B and C are matrices whose sizes are such that the operations are well defined and k is a scalar and n is a positive integer.
I_{n} is the identity matrix of size n × n whose diagonal entries are all equal to 1 and all non diagonal entries equal to zero.
0 is the zero matrix whose entries are all zeros.
Page Content
 Properties of Matrix Addition
 Properties of Matrix Multiplication
 Distributive Properties of Matrices
 Properties of Matrix Multiplication by Scalars
 Properties of Matrix Transpose
 Properties of Matrix Inverse
 References
Properties of Matrix Addition
 A + 0 = 0 + A = A , where 0 is the zero matrix.
 A + B = B + A Commutativity of Addition of Matrices
 (A + B) + C = A + (B + C) Associativity of Addition of Matrices
Properties of Matrix Multiplication
 A ( B C ) = ( A B ) C Associativity of Multiplication of Matrices
 A I_{n} = A , where I_{n} is the identity matrix.
 I_{n} A = A
 0 A = 0 , where 0 is the zero matrix.
 Note that in general AB ≠ BA
Distributive Properties of Matrices
 A(B ~+mn~ C) = AB ~+mn~ AC
 (A ~+mn~ B)C = AC ~+mn~ BC
Properties of Matrix Multiplication by Scalars
 k(A ~+mn~ B) = k A ~+mn~ k B
 (k ~+mn~ m)A = k A ~+mn~ m A
 k(m A) = (k m)A
 k(A B) = (k A)B = A(k B)
Properties of Matrix Transpose
In what follows, A^{T} is the transpose matrix.
 (A^{T})^{T} = A
 (A ~+mn~ B )^{T}) = A^{T} ~+mn~ B^{T}
 (k A)^{T} = k A^{T}
 (A B)^{T} = B^{T} A^{T}
 (I_{n} )^{T} = I_{n}
Properties of Matrix Inverse
In what follows, A^{1} is the inverse matrix .
More References and links

Linear Algebra  Questions with Solutions

Matrices with Examples and Questions with Solutions.

Determinant of a Square Matrix.

Inverse Matrix Questions with Solutions.

Transpose of a Matrix.

orthogonal matrix

orthonormal
 Elementary Linear Algebra  7 th Edition  Howard Anton and Chris Rorres
 Introduction to Linear Algebra  Fifth Edition (2016)  Gilbert Strang
 Linear Algebra Done Right  third edition, 2015  Sheldon Axler
 Linear Algebra with Applications  2012  Gareth Williams