1. For any matrix A there exists the opposite matrix (-A) such that the sum

A + (- A) = A - A = 0

is a zero-matrix.

2. If two matrices, A and B, have the same size then

A + B = B + A.

3. If three matrices, A, B, and C, have the same size then

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

4. The transpose of the sum of matrices is the sum of transpose of the matrices:

.

All Addition Properties follow from the properties of real numbers. The proofs are left to the reader.

Properties Involving Multiplication

proof

1. Let A be a matrix, and let and be numbers. Then

.

2. Let be a number and let A and B be two matrices such that the product AB is defined. Then

.

3. The product of a matrix A and the zero-matrix 0 is the zero-matrix:

.

4. Let A, B, and C be three matrices such that all multiplications are appropriate. Then

(AB)C = A(BC).

5. Let A and B be two matrices such that the product AB is defined. Then

.

6. Let A be a matrix and let E be the identity matrix of the n-th order. Then

AE = A.

In a special case when A is a square matrix:

AE = EA =A.

Properties Involving Addition and Multiplication

proof

1. Let A, B, and C be three matrices such that the corresponding products and sums are defined. Then

A(B + C) = AB + AC

and

(A + B)C = AC + BC.

2. Let be a number and let A and B be two matrices of the same size. Then

.

Note: By the definitions and properties of matrices, a number is a particular case of a matrix of the first order. Therefore, a set of real numerical matrices is a generalization of the set of real numbers.