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.

- 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

- 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

- 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

- A(B ± C) = AB ± AC
- (A ± B)C = AC ± BC

- k(A ± B) = k A ± k B
- (k ± m)A = k A ± m A
- k(m A) = (k m)A
- k(A B) = (k A)B = A(k B)

In what follows, A^{T} is the transpose matrix.

- (A
^{T})^{T}= A - (A ± B )
^{T}) = A^{T}± B^{T} - (k A)
^{T}= k A^{T} - (A B)
^{T}= B^{T}A^{T} - (I
_{n})^{T}= I_{n}

In what follows, A^{-1} is the inverse matrix .

- 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