From ControlTheoryPro.com

Contents
1 Matrix Operations
1.1 Multiplication
In Linear Algebra 2 matrices can be multiplied by each only if the inner dimensions are equal.
 where
 is a matrix of n rows and m columns
 is a matrix of m rows and o columns
 then must be a matrix of n rows and o columns.
Notice that C has the same number of rows (m) and B does columns  the inner dimensions of the product. This also implies that the order of multiplication is important.
To multiply 2 matrices in MATLAB use * operator
>> a = [1, 2 ; 3, 4] ans = 1 2 3 4 >> b = [3, 4 ; 5, 6]; >> a * b ans = 13 16 29 36 >> c = [3, 4, 5 ; 6, 7, 8]; >> a * c ans = 15 18 21 33 40 47
1.2 Division
Division is more constrained than multiplication. In Algebra division is simple
In Linear Algebra the order of the multiplication must be preserved. So
 and

 where
 is the identity matrix.
 where
In order to do division the matrix "in the denominator" must be invertable. Not all matrices are invertable. First and foremost the matrix must be square since For those matrices that are not invertable the pseudoinverse can be used. In MATLAB the pseudoinverse can be found using the pinv command
>> a = [1, 2 ; 3, 4]; >> a^1 ans = 2.0000 1.0000 1.5000 0.5000 >> pinv(a) ans = 2.0000 1.0000 1.5000 0.5000
 These 2 operations lead to the same results since a is invertable. However, c is not
>> c = [3, 4, 5 ; 6, 7, 8]; >> c^1 ??? Error using ==> mpower Matrix must be square. >> pinv(c) ans = 1.2778 0.7778 0.1111 0.1111 1.0556 0.5556
See this sie on pseudoinverses for more information.
'Pseudoinverse' 
1.3 Squaring a matrix
Remember that squaring any value is the same as
In order to square a matrix in Linear Algebra the matrix must be square (same number of rows and columns) so that the inner dimensions are equal. Given a square matrix you can then take that matrix to some power in MATLAB using the ^ operator. For example
>> a = [1, 2 ; 3, 4]; >> a^2 ans = 7 10 15 22
If, however, you wish to square the value in each element then use the . and ^ operators together
>> a.^2 ans = 1 4 9 16
The period before the ^ operator tells MATLAB to perform the operation element by element. The . operator works with division, multiplication, and other operators.
1.4 Determinant
To get the determinant of a matrix use the det command on that matrix, as follows
>> a = [1, 2 ; 3, 4]; >> det(a) ans = 2
1.5 Transpose
The transpose of a matrix is that matrix with the rows and columns swapped. To get the transpose of a matrix use the transpose command or the ' operator.
>> a = [1, 2 ; 3, 4] ans = 1 2 3 4 >> transpose(a) ans = 1 3 2 4
 or
>> a' ans = 1 3 2 4