# 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

Evaluate (AB)C and A(BC). Check that you get the same matrix.

so

On the other hand, we have

so

Example. Consider the matrices

It is easy to check that

and

These two formulas are called linear combinations. More on linear combinations will be discussed on a different page.

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

In particular, we have

The matrix In has similar behavior as the number 1. Indeed, for any nxn matrix A, we have

A In = In A = A

The matrix In is called the Identity Matrix of order n.

Example. Consider the matrices

Then it is easy to check that

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

In particular, we have

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Author: M.A. Khamsi