MATH 323 Lecture 4

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Matrices (cont'd)


1 and 2 are called a ring because they have the + and · operations defined on them since they are square matrices.

Multiplication (cont'd)

NOT commutative:

Due to associativity of multiplication in #Theorem 1.3.2, we do not write parentheses around multiplication:


for integer

Identity Matrix

Plays the role of "1" in multiplication:

, where

(In other words, 1s along diagonal, 0s everywhere else) ...such that

Standard Basis

Acts as an identity vector:

, where is a vector of 0s, except the th element is 1.

Zero Matrix

Plays role of 0 in addition:

...such that

Inverse Matrix

If for matrices and , we say than is the inverse of : .

Not every matrix is invertible. A matrix that does not have a multiplicative inverse is said to be singular.

Transpose of a Matrix

For a matrix , the transpose of , written , will be and is defined as follows:

, where

Geometrically, the first row becomes the first column, second row becomes second column, etc. The matrix is simply "reflected" about its "diagonal"


  1. ,
  2. (opposite order)

A matrix is said to be symmetric iff . This means that


Diagonal Matrix

A diagonal matrix is a square () matrix that has values along its diagonal ()

A special form of diagonal matrices is when (i.e. all numbers along diagonal are the same): these special diagonal matrices commute with arbitrary matrices

Matrices and Graphs

A graph consists of vertices (data points) and edges that connect them.

A graph with entries ( through ) can be represented by a adjacency matrix:

First, let be the (, ) entry of .

represents the number of walks of length from to .

Theorem 1.3.2

For all and for all , the indicated operations are defined:

  1. (commutativity of addition)
  2. (associativity of addition)
  3. (associativity of multiplication)
  4. (right distributivity)
  5. (left distributivity)

Theorem 1.3.3

If and are nonsingular matrices, then is also nonsingular and

(note the opposite order on the right-hand side)


For nonsingular matrices , is also nonsingular and