Laws of Matrix Algebra
6 Old Rules, 5 Still Apply
We had associative, commutative and distributive laws for scalar algebra, we can think of them as the six bullet points below. Only the multiplicative-commutative law no longer works for matrix, the other rules work for matrix as well as scalar algebra.
Associative laws work as in scalar algebra for matrix
Commutative Law works as well for addition
- with scalars, we know
, but commutative law for matrix multiplication does not work, Matrix 
. The matrix dimensions might not even match up for multiplication. (see below for examples)
And Distributive Law still applies to matrix
Example for 
% Non-Square
A = rand(2,3)
B = rand(3,4)
% This is OK
disp(A*B)
% This does not work
try
B*A
catch ME
disp('does not work! Dimension mismatch')
end
% Square
A = rand(3,3)
B = rand(3,3)
% This is OK
A*B
% This works, but result differs from A*B
B*A
4 New Rules for Transpose
In scalar algebra, transpose does not make sense. Given matrix A, 
is the transpose matrix of A where each row of A becomes columns in . If A is M by N, then is N by M.
is the transpose matrix of A where each row of A becomes columns in . If A is M by N, then is N by M.Given matrix A and scalar value r:
- 1:
- 2:
- 3:
- 4:
For the 4th rule, suppose matrix A is has L rows and M columns, and the matrix B has M rows and Ncolumns. 
is a L by N matrix, 
is a N by L matrix. This is equal to 
, where we have a N by M matrix 
multiplied by a M by L matrix , and the resulting matrix is N by L.
is a L by N matrix, is a N by L matrix. This is equal to , where we have a N by M matrix multiplied by a M by L matrix , and the resulting matrix is N by L. A = rand(2,3)
Atranspose = (A')