As pointed out above, sometimes it is possible to multiply matrices in one order but not in the other order. What if it makes sense to multiply them in either order? Will they be equal then?
Example 4.1.1 Compare
The first product is

the second product is

and you see these are not equal. Therefore, you cannot conclude that AB = BA for matrix multiplication. However, there are some properties which do hold.
Proposition 4.1.2 If all multiplications and additions make sense, the following hold for matrices, A,B,C and a,b scalars.
 (4.9) 
 (4.10) 
 (4.11) 
Proof: Using the above definition of matrix multiplication,
Consider 4.11, the associative law of multiplication. Before reading this, review the definition of matrix multiplication in terms of entries of the matrices.
Another important operation on matrices is that of taking the transpose. The following example shows what is meant by this operation, denoted by placing a T as an exponent on the matrix.

What happened? The first column became the first row and the second column became the second row. Thus the 3 × 2 matrix became a 2 × 3 matrix. The number 3 was in the second row and the first column and it ended up in the first row and second column. This motivates the following definition of the transpose of a matrix.
Definition 4.1.3 Let A be an m × n matrix. Then A^{T} denotes the n × m matrix which is defined as follows.

The transpose of a matrix has the following important property.
Lemma 4.1.4 Let A be an m × n matrix and let B be a n × p matrix. Then
 (4.12) 
and if α and β are scalars,
 (4.13) 
Proof: From the definition,
Definition 4.1.5 An n × n matrix A is said to be symmetric if A = A^{T}. It is said to be skew symmetric if A^{T} = −A.
Example 4.1.6 Let

Then A is symmetric.
Example 4.1.7 Let

Then A is skew symmetric.
There is a special matrix called I and defined by I_{ij} = δ_{ij} where δ_{ij} is the Kronecker symbol defined by

It is called the identity matrix because it is a multiplicative identity in the following sense.
Lemma 4.1.8 Suppose A is an m×n matrix and I_{n} is the n×n identity matrix. Then AI_{n} = A. If I_{m} is the m × m identity matrix, it also follows that I_{m}A = A.
Proof:

and so AI_{n} = A. The other case is left as an exercise for you.
Definition 4.1.9 An n×n matrix A has an inverse A^{−1} if and only if there exists a matrix, denoted as A^{−1} such that AA^{−1} = A^{−1}A = I where I =

If it acts like an inverse, then it is the inverse. This is the message of the following proposition.
Proof: From the definition, B is an inverse for A. Could there be another one B^{′}?

Thus, the inverse, if it exists, is unique. ■