# Matrices

**4.4. Special Matrices.**

1. A ** square matrix ** is a matrix with the same number of rows as columns.

2. A **diagonal matrix** is a square matrix whose entries off the main
diagonal are

zero.

3. An **upper triangular matrix** is a matrix having all the entries below
the main

diagonal equal to zero.

4. A ** lower triangular matrix** is a matrix having the entries above the
main diagonal

equal to zero.

5. The n × n** identity matrix**, I, is the n × n matrix with ones down the
diagonal

and zeros elsewhere .

6. The **inverse** of a square matrix, A, is the matrix A^{-1}, if it exists,
such that

AA^{-1} = A^{-1}A = I.

7. The **transpose** of a matrix
is
.

8. A **symmetric matrix** is one that is equal to its transpose.

Discussion

Many matrices have special forms and special properties . Notice that, although

a diagonal matrix must be square, no such condition is put on upper and lower

triangular matrices.

The following matrix is a diagonal matrix (it is also upper and lower
triangular ).

The following matrix is upper triangular.

The next matrix is the transpose of the previous matrix .
Notice that it is lower

triangular.

The identity matrix is a special matrix that is the
multiplicative identity for any

matrix multiplication. Another way to define the identity matrix is the square
matrix

where if i
≠ j and . The n × n identity I has the
property

that IA = A and AI = A, whenever either is defined. For example,

The inverse of a matrix A is a special matrix A^{-1} such
that AA^{-1} = A^{-1}A = I. A

matrix must be square to define the inverse. Moreover, the inverse of a matrix
does

not always exist.

Example 4.4.1.

so that

The transpose of a matrix is the matrix obtained by
interchanging the rows for

the columns. For example, the transpose of

If the transpose is the same as the original matrix, then
the matrix is called

symmetric. Notice a matrix must be square in order to be symmetric .

We will show here that matrix multiplication is distributive over matrix
addition.

Let
and
be m × n matrices and
let be an n × p

matrix. We use the definitions of addition and matrix multiplication and the
distributive

properties of the real numbers to show the distributive property of matrix

multiplication. Let i and j be integers with 1≤ i ≤ m and 1 ≤ j ≤ p. Then the

element in the i-th row and the j-th column in (A + B)C would be given by

This last part corresponds to the form the element in the
i-th row and j-th column

of AC + BC. Thus the element in the i-th row and j-th column of (A + B)C is the

same as the corresponding element of AC + BC. Since i and j were arbitrary this

shows (A + B)C = AC + BC.

The proof that C(A+B) = CA+CB is similar. Notice that we must be careful,

though, of the order of the multiplication. Matrix multiplication is not
commutative.

**4.5. Boolean Arithmetic .** If a and b are binary digits (0 or 1), then

Definitions 4.5.1. Let A and B be n × m matrices.

1. The** meet** of A and B:

2. The** join** of A and B:

Definition 4.5.1. Let be
m × k and be k × n. The
**Boolean
product** of A and B, , is the m × n matrix
defined by

Discussion

Boolean operations on zero -one matrices is completely analogous to the standard

operations, except we use the Boolean operators
and on the binary digits
instead

of ordinary multiplication and addition , respectively.

**4.6. Example 4.6.1.**

Example 4.6.1. Let

Then

Here are more details of the Boolean product in Example 4.6.1:

Exercise 4.6.1.

Find

Exercise 4.6.2.

Find

Exercise 4.6.3.

Find , the Boolean
product of A with itself n times . Hint: Do exercise 4.6.2

first.

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