Recall the following definition of matrix multiplication and addition.
Definition A.2.1 Letting M_{ij} denote the entry of M in the i^{th} row and j^{th} column, let A be an m × n matrix and let B be an n × p matrix. Then AB is an m × p matrix and

If A,B are both m × n, then
Recall the following properties of matrix arithmetic which follow right away from the above.
 (1.4) 
the commutative law of addition,
 (1.5) 
the associative law for addition,
 (1.6) 
the existence of an additive identity,
 (1.7) 
the existence of an additive inverse. Also, for α,β scalars, the following also hold.
 (1.8) 
 (1.9) 
 (1.10) 
 (1.11) 
The above properties, 1.41.11 are known as the vector space axioms and the fact that the m × n matrices satisfy these axioms is what is meant by saying this set of matrices with addition and scalar multiplication as defined above forms a vector space.
There are also properties which are related to matrix multiplication.
Proposition A.2.2 If all multiplications and additions make sense, the following hold for matrices, A,B,C and a,b scalars.
 (1.12) 
 (1.13) 
 (1.14) 
Proof: Using the above definition of matrix multiplication,


showing that A
Consider 1.14, the associative law of multiplication. Before reading this, review the definition of matrix multiplication in terms of entries of the matrices.
Recall also that AB is sometimes not equal to BA. For example,

Definition A.2.3 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 A.2.4 Let A be an m × n matrix and let B be a n × p matrix. Then
 (1.15) 
and if α and β are scalars,
 (1.16) 
Proof: From the definition,

1.16 is left as an exercise. ■
Definition A.2.5 A real n × n matrix A is said to be symmetric if A = A^{T}. It is said to be skew symmetric if A^{T} = −A.There is a special matrix called I and defined by

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.
The following lemma follows from the above definition.
Lemma A.2.6 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.
Definition A.2.7 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. ■
Recall the definition of the special vectors e_{k}.

where the 1 is in the k^{th} position from the left.
Definition A.2.9 Let A be a matrix. N
There is a fundamental result in the case where m < n. In this case, the matrix A looks like the following.
Proof: It is clear that the theorem is true if A is 1 × n with n > 1. Suppose it is true whenever A is m − 1 × k for m − 1 < k. Say A is m × n with m < n,m > 1,

If a_{1} = 0, consider the vector x = e_{1}. Ae_{1} = 0. If a_{1}≠0, do row operations to obtain a matrix Â such that the solutions of Ay = 0 are the same as the solutions of Ây = 0, and Â =

Now B is m − 1 × n − 1 where m − 1 < n − 1 and so by induction, there is a nonzero vector x such that Bx = 0. Consider the nonzero vector z =