A * matrix* is a rectangular array of numbers:

Actually, the entries can be more general than numbers, but you can
think of the entries as numbers to start. I'll give a rapid account
of basic matrix arithmetic; you can find out more in a course in * linear algebra*.

* Definition.* (a) If A is a matrix,
is the element in the row and column.

(b) If a matrix has m rows and n columns, it is said to be an matrix, and m and n are the *
dimensions*. A matrix with the same number of rows and columns is
a * square matrix*.

(c) If A and B are matrices, then if their corresponding entries are equal; that is, if for all i and j.

Note that matrices of different dimensions can't be equal.

(d) If A and B are matrices, their *
sum* is the matrix obtained by adding corresponding
entries of A and B:

(e) If A is a matrix and k is a number, the *
product* of A by k is the matrix obtained by multiplying the
entries of A by k:

(f) The * zero matrix* is the
matrix all of whose entries are 0.

(g) If A is a matrix, then is the matrix obtained by negating the entries of A.

(h) If A and B are matrices, their *
difference* is

* Example.* Suppose

(a) What are the dimensions of A?

(b) What is ?

(a) A is a matrix: It has 2 rows and 3 columns.

(b) is the element in row 2 and column 1, so .

The following results are unsurprising, in the sense that things work the way you'd expect them to given your experience with numbers. (This is not always the case with matrix multiplication, which I'll discuss later.)

* Proposition.* Suppose p and q are numbers and
A, B, and C are matrices.

(a) (Associativity) .

(b) (Commutativity) .

(c) (Zero Matrix) If 0 denotes the zero matrix, then

(d) (Additive Inverses) If 0 denotes the zero matrix, then

(e) (Distributivity)

* Proof.* Matrix equality says that two matrices
are equal if their corresponding entries are equal. So the proofs of
these results amount to considering the entries of the matrices on
the left and right sides of the equations.

By way of example, I'll prove (b). I must show that and have the same entries.

The first and third equalities used the definition of matrix addition. The second equality used commutativity of addition for numbers.

Matrix multiplication is more complicated. Your first thought might
be to multiply two matrices the way you add matrices --- that is, by
multiplying corresponding entries. That is *not* how matrices
are multiplied, and the reason is that it isn't that useful. What is
useful is a more complicated definition which uses dot products.

Suppose A is an matrix and B is an matrix. Note that the number of columns of A must equal the number of rows of B. To form the matrix product , I have to tell you what the entry of is. Here is the description in words: is the dot product of the row of A and the column of B.

The resulting matrix will be an matrix.

This is best illustrated by examples.

* Example.* Compute the matrix product

This is the product of a matrix and a matrix. The product should be a matrix. I'll show the computation of the entries in the product one-by-one. For each entry in the product, I take the dot product of a row of the first matrix and a column of the second matrix.

:

:

:

:

:

:

Thus,

* Example.* Multiply the following matrices:

* Example.* Multiply the following matrices:

Note that this is essentially the dot product of two 4-dimensional vectors.

* Example.* Multiply the following matrices:

The formal definition of matrix multiplication involves summation notation. Suppose A is an matrix and B is an matrix, so the product makes sense. To tell what is, I have to say what a typical entry of the matrix is. Here's the definition:

Let's relate this to the concrete description I gave above. The summation variable is k. In , this is the column index. So with the row index i fixed, I'm running through the columns from to . This means that I'm running down the row of A.

Likewise, in the variable k is the row index. So with the column index j fixed, I'm running through the rows from to . This means that I'm running down the column of B.

Since I'm forming products and then adding them up, this means that I'm taking the dot product of the row of A and the column of B, as I described earlier.

Proofs of matrix multiplication properties involve this summation definition, and as a consequence they are often a bit messy with lots of subscripts flying around. I'll let you see them in a linear algebra course.

* Definition.* The * identity matrix* is the matrix with 1's down the
* main diagonal* (the diagonal going from upper
left to lower right) and 0's elsewhere.

For instance, the identity matrix is

* Proposition.* Suppose A, B, and C are matrices
(with dimensions compatible for multiplication in all the products
below), and let k be a number.

(a) (Associativity) .

(b) (Identity) and (where I denotes an identity matrix compatible for multiplication in the respective products).

(c) (Zero) and (where 0 denotes a zero matrix compatible for multiplication in the respective products).

(d) (Scalars) .

(e) (Distributivity)

I'll omit the proofs, which are routine but a bit messy (as they involve the summation definition of matrix multiplication).

Note that commutativity of multiplication is not listed as a
property. In fact, it's false --- and it one of a number of ways in
which matrix multiplication *does not* behave in ways you
might expect. *It's important to make a note of things which
behave in unexpected ways.*

* Example.* Give specific matrices A and B such that .

There are many examples. For instance,

This shows that matrix multiplication is not commutative.

* Example.* Give specific matrices A and B such that , , but .

There are many possibilities. For instance,

* Example.* Give specific nonzero matrices A. B, and C such that but .

There are lots of examples. For instance,

But

This example shows that you can't "divide" or "cancel" A from both sides of the equation. It works in some cases, but not in all cases.

* Definition.* Let A be an matrix. The * inverse* of A is a
matrix which satisfies

(I is the identity matrix.) A matrix which has an
inverse is * invertible*.

* Remark.* (a) If a matrix has an inverse, it is
unique.

(b) Not every matrix has an inverse. *
Determinants* provide a criterion for telling whether a matrix is
invertible: An real matrix is invertible if and only if
its determinant is nonzero.

* Proposition.* Suppose A and B are invertible
matrices. Then:

(a) .

(b) .

* Proof.* I'll prove (b).

This shows that is the inverse of , because it multiplies to the identity matrix in either order.

In general, the most efficient way to find the inverse of a matrix is
to use * row reduction* (*
Guassian elimination*), which you will learn about in a linear
algebra course. But we can give an easy formula for matrices.

* Proposition.* Consider the real matrix

Suppose . Then

* Proof.* Just compute:

You can check that you also get the identity if you multiply in the opposite order.

* Example.* Find the inverse of .

* Example.* Use matrix inversion to solve the
system of equations:

You can write the system in matrix form:

(Multiply out the left side for yourself and see that you get the original two equations.) Now the inverse of the matrix is

Multiply the matrix equation above *on the left of both sides*
by the inverse:

That is, and .

Copyright 2017 by Bruce Ikenaga