Algebraic Properties of Matrix Operations

Algebraic Properties of Matrix Operations

In this page, we give some general results about the three operations: addition, multiplication, and multiplication with numbers, called scalar multiplication.

From now on, we will not write (mxn) but mxn.
 

Properties involving Addition. Let A, B, and C be mxn matrices. We have

1.

A+B = B+A

2.

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

3.

where is the mxn zero-matrix (all its entries are equal to 0);

4.

if and only if B = -A.

Properties involving Multiplication.

1.

Let A, B, and C be three matrices. If you can perform the products AB, (AB)C, BC, and A(BC), then we have
 

 

(AB)C = A (BC)

 

 

Note, for example, that if A is 2x3, B is 3x3, and C is 3x1, then the above products are possible (in this case, (AB)C is 2x1 matrix).

2.

If and are numbers, and A is a matrix, then we have
 

 

 

 

3.

If is a number, and A and B are two matrices such that the product is possible, then we have
 

 

 

 

4.

If A is an nxm matrix and the mxk zero-matrix, then
 

 

 

 

Note that is the nxk zero-matrix. So if n is different from m, the two zero-matrices are different.

Properties involving Addition and Multiplication.

1.

Let A, B, and C be three matrices. If you can perform the appropriate products, then we have
 

 

(A+B)C = AC + BC

 

 

and
 

 

A(B+C) = AB + AC

 

 

2.

If and are numbers, A and B are matrices, then we have
 

 

 

 

and
 

 

 

 

Example. Consider the matrices
 

Answer. We have
 

Example. Consider the matrices
 

We have seen that matrix multiplication is different from normal multiplication (between numbers). Are there some similarities? For example, is there a matrix which plays a similar role as the number 1? The answer is yes. Indeed, consider the nxn matrix
 

Example. Consider the matrices
 

The identity matrix behaves like the number 1 not only among the matrices of the form nxn. Indeed, for any nxm matrix A, we have