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 zeromatrix (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 zeromatrix, then
Note that
is the nxk zeromatrix. So if n is different from m, the two zeromatrices 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.
Answer. We have
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 I_{n} has similar behavior as the number 1. Indeed, for any nxn matrix A, we have
A I_{n} = I_{n} A = A
The matrix I_{n} 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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