**1 Terminology**

A matrix is a rectangular array of numbers, for example

,or

The numbers in any matrix are called its entries. The
entries of a matrix are organized into rows and

columns, which are simply the horizontal and vertical (resp.) lists of entries
appearing in the matrix. For

example, if

then the rows of M are
and whereas the

columns of M are

,and
.

It is worth noting that an m · n matrix will have m rows
with n entries each, and n columns with m entries

each. That is, the number of entries in any row of a matrix is the number of
columns of that matrix, and

vice versa. This is readily apparent in each of the examples above.

The dimensions of a matrix are the numbers of rows and columns it has. If a
matrix has m rows and n

columns we say that it is an m · n matrix (note that we always list the number
of rows first). So, the first

four matrices above have dimensions
and , respectively. The dimensions of the
matrix

M are An m · n matrix is called square if m
= n. Thus, the only example of a square matrix above

is the second.

So that we can more easily refer to various entries in matrices, we index the
columns of a matrix from

left to right and the rows from top to bottom. For example, the first column of
M (above) is

the second column is

the third column is

etc. The first row of M is
the second row is
and the third row is

We can use this numbering scheme to easily
refer to entries in a matrix: we call the

entry located in row i and column j the i, j-entry. For the matrix

the 1, 1-entry is 1, the 3, 2-entry is -1, the 4, 3-entry
is 9 and the 2, 1-entry is -1/6.

To write down a matrix with variable entries we use variables with subscripts
that indicate their position

in the matrix, using the convention described above. A generic m· n matrix can
therefore be denoted

or just for
short.

An m ·1 matrix has the form

and is called, appropriately, a column vector. Notice that
since a column vector has only a single column we

have used only single subscripts to index its entries. Likewise, a 1· n matrix
looks like

and is called a row vector. When we use the word vector
with no qualification we will usually mean a column

vector. Column vectors give us another shorthand for writing down generic
matrices. Notice that if we use

the matrix A in (1) and set

(i.e. we use the entries in the j-th column of A as the
entries in a_{j}) then we can write

In a similar way one can also use the rows of A to ex press
A in terms of row vectors, but since we won't be

using this idea later we won't bother to write it out.

**2 Scalar multiplication and addition of matrices **

Having dispensed with the basic termino logy and notation of matrices, we now
turn to how they are ma-

nipulated algebraically . We will see that it is possible to add, subtract and
multiply matrices together, but

only if certain restrictions on their dimensions are met. We begin with the
notion of scalar multiplication.

Given an m · n matrix A = (a_{ij}) and a number (scalar) c we define

That is, cA is the matrix obtained by multiplying every
entry of A by c. As examples, if

then

Adding two matrices is also done entry-by-entry. If A =
(a_{ij}) and B = (b_{ij}) are two m· n matrices, then

their sum is A + B = (a_{ij} +b_{ij} ). That is, the i, j-entry of A+ B is the sum of the i, j-entries of A and B. It

is important to note that is is only possible to add two matrices if they have
exactly the same dimensions.

Here's an example: if

then

and

The fol lowing theorem summarizes the main · properties of
matrix addition. The proofs of these properties

follow directly from the definitions made so far and are left to the reader. We
will find it useful to be able

to refer to the m· n zero matrix , which is the matrix all of whose entries are
zero.

**Theorem ·1.** Let A, B and C be m· n matrices, let c be a real number and let
**0 **denote the m· n zero

matrix. Then

**3 Matrix multiplication**

Defining the matrix product is a two step process . First we will define what it
means to multiply a matrix

by a column vector and then we'll use that to tell us how two multiply matrices
in general. Let A be an

m ·n matrix and let v be an· n ·1 column vector (notice that the vector v has as
many entries as A has

columns). Write A in terms of its columns as above,

and write out the entries of v as

The product of A with v is defined to be

In words, we multiply the columns of A by the respective
entries of v and then add the results together.

According to this definition, the product of an m· n matrix and an· n ·1 column
vector is an m ·1 column

vector, i.e. the product is a column with as many entries as A has rows.

The process of multiplying a matrix by a vector is straightforward enough once
one is used to the

definition. Let's look at some examples. Suppose that we take

The matrix A can only be multiplied by column vectors with
2 entries while B can only be multiplied by by

column vectors with 4 entries. So, if we take

then

and

Since we can· now multiply matrices by (suitably sized)
column vectors, we can develop a way to multiply

matrices by other (suitably sized) matrices. Let A be an m· n matrix and let B be
a n ·p matrix. Notice

that B has as many rows as A has columns. In ·particular, the columns of B are n
·1 column vectors and

can therefore in dividually be multiplied by A. To be more specific, write B in
terms of its columns:

where each b_{j} is an· n ·1 column vector. We define the
product of A and B to be

That is, to multiply two matrices simply multiply the first
matrix by the columns of the second and use the

results as the columns in a new matrix. Since each A_{j} is an m ·1 column vector,
and there are exactly p of

them, we find that AB is an m · p matrix.

Let's look at a quick example. Take

The product AB makes sense since A has as many columns as
B has rows. The definition of matrix

multiplication says that

We find that

and

so that

The n· n identity matrix I is the (square) matrix all of
whose entries are zero except for those along the

\main diagonal" which are all equal to 1. Symbolically

The and
identity matrices are then

respectively.

The following theorem gives the main ·properties of matrix multiplication. These
all follow directly

from the definitions, but some are harder to prove than others, most notably that
matrix multiplication is

associative.

**Theorem 2.** Let A be m· n, B and C be n ·p, D be p· q, and let c be a real number.
Then

1. A(B + C) = AB + AC,

2. (B + C)D = BD + CD,

3. (AB)D = A(BD),

4. if I is the m m identity matrix then IA = A,

5. if I is the n· n identity matrix then AI = A,

6. c(AB) = (cA)B = A(cB).

**4 Exercises**

In exercises 1 and 2, let

and compute each matrix sum or product if it is defined. If
it is not defined, explain why.

**Exercise 1.**

a. A - B

b. A - 3E

c. 2A + DB

d. AC

**Exercise 2.**

a. A + CB

b. 3BC - A

c. CAD

d. CA - E

**Exercise 3.** If show
that AB ≠ BA but that

AC = CA.

**Exercise 4.** If construct a nonzero
matrix
B (with two distinct columns) so that

AB is the zero matrix.

**Exercise 5.** If A is an· n n matrix, we say the n· n matrix B is the inverse of A
if AB = BA = I,

where I is the n· n identity matrix. Show that if
with then the inverse of A is

.

**Exercise 6. **If and
, use the inverse of A (see the previous
exercise) to solve

the matrix equation Ax = b for x.

**Exercise 7.** If find a nonzero vector v so
that Av = 0.