Properties
A. Equality.
-
Matrix are equal only when they are of the same order, and each cell entry in one matrix equals the cell entry in the same row and same column of the other matrix. So, we can see that:
is the same as: 
while neither of these matrices are equal to 
.
B. 
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Notice that the transpose of the transpose of a matrix equals the original matrix. For example, if
then 
and 
.
C. Transpose of (ABC)
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If we consider the expression:
. A little bit of exercise should convince you, for example, that: 
D. Transpose of Symmetric Matrices
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All symmetric matrices (and only symmetric matrices) are equal to their transposes. I.e.,
.
E. Distribution Law
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Matrix addition (and subtraction) obey the distribution law; i.e., A(B+D)=AB+AD; A(B-C)=AB-AC; and (A+B)C=AC+BC.
F. Commutative Law
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Matrix addition obeys the commutative law, i.e., A+B=B+A
G. Associate Law
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Matrix addition and subtraction obey the associative law; i.e., A+(B+C)=(A+B)+C and A-(B-C)=(A-B)-C.
H. Order of a Product
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Notice that, when multiplying two matrices together that the number of rows in the resulting product matrix is equal to the number of rows in the first matrix, and the number of columns in the resulting product matrix is equal to the number of columns in the second matrix.
I. Multiplication by Identity
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The product of a matrix, say A, with a suitable identity matrix equals the original matrix. E.g.:
(Try this out and make sure you believe it.) Similarly, if we premultiply by a suitable matrix, 
. Why, based on discussions elsewhere in this document did we need to have a 3 x 3 identity matrix in the first example and a 4 x 4 identity matrix in the second?
J. Product of Diagonal Matrices
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The product of diagonal matrices, say D and E, is a third diagonal matrix, say F, with diagonal entries equal to the products of corresponding entries in the first two diagonal matrices. F=DE means in this case that:
for across i=1 to k, where k is the number of rows or columns.
K. Order of Product of Diagonal Matrices
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The product of two diagonal matrices is invariant over reversal in the order of multiplication. Let D, E, and F be diagonal matrices where: DE=F. then ED=F as well.
L. Premultiplication by Diagonal Matrix
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Premultiplication of a matrix, say A, by a diagonal matrix, say D, multiplies the entries in each row of A by the diagonal entry in D in the diagonal cell corresponding to that row of A. B=DA means that
. So, for example, we can see that if 
and 
then DA=
M. Postmultiplication by Diagonal
-
Postmultiplication of a matrix, say A, by a diagonal matrix, say D, multiplies the entries in each column of A by the diagonal entry in D in the diagonal cells corresponding to that column of A. So, if we consider B=AD and we consider A=
and we now consider 
then AD=
N. Matrix Addition as Multiplication
1. There is a special relationship between matrix addition and multiplication. Basically, any matrix addition problem can be turned into one of multiplication, by adding on suitable identity matrices at the appropriate places. Generally, if there exist two matrices 
and another matrix B of the same order, then the sum of A and B may be written as: 
(Where the the identity matrices have the dimension indicated and the vertical and horizontal lines are merely indicating that we're adjoining two matrices. So, for example, consider: 
and 
. Now we could, of course, simply write out: 
but that would be making things too simple. We can also write this as: 
Now why, you may well ask, would anyon in their right mind rewrite a simple addition problem as a multiplication problem? The answer to this has to do with the fact that statisticians doing covariance structure analysis like to express all arrows and paths as a multiplication problem of the form: 
or, more generally, as: 
. Basically, this helps us to express a path model as a prodicted variance/covariance or correlation matrix, and also helps us to see connections between the statistical models for structural equations proposed by different statisticians.