**Read** Sections 3.1 and 3.2 for next class.
Work through recommended homework questions.

**Tutorials:** No quizzes this week, focused on review.

**Help Centers:** Monday-Friday 2:30-6:30 in MC 106.

These **lecture notes** now available in pdf format as well,
a day or two after each lecture.
Be sure to let me know of technical problems.

We also aren't covering Section 2.5.

**Definition:** A **matrix** is a rectangular array of numbers called the **entries**.
The entries are usually real (from $\R$), but may also be complex (from $\C$) or be from $\Z_m$.

**Examples:**
$$
\kern-8ex
% The Rules create some space below the matrices:
\mystack{
A = \bmat{ccc}
1 & -3/2 & \pi \\
\sqrt{2} & 2.3 & 0
\emat
\Rule{0pt}{0pt}{18pt}
}{2 \times 3}
\qquad
\mystack{
\bmat{rr}
1 & 2 \\ 3 & 4
\emat
\Rule{0pt}{0pt}{22pt}
}{\mystack{\strut 2 \times 2}{\textbf{square}}}
\qquad
\mystack{
\bmat{rrrr}
1 & 2 & 3 & 4
\emat
\Rule{0pt}{0pt}{30pt}
}{\mystackthree{1 \times 4}{\textbf{row matrix}}{\text{or }\textbf{row vector}}}
\qquad
\mystack{
\bmat{r}
1 \\ 2 \\ 3 \\ 4
\emat
\Rule{0pt}{0pt}{30pt}
}{\mystackthree{4 \times 1}{\textbf{column matrix}}{\text{or }\textbf{column vector}}}
$$
The entry in the $i$th row and $j$th column of $A$ is usually written $a_{ij}$
or sometimes $A_{ij}$. For example,
$$A_{11} = 1, \quad A_{23} = 0, \quad A_{32} \text{ doesn't make sense} . $$

**Definition:** An $m \times n$ matrix $A$ is **square** if $m = n$.
The **diagonal entries** are $a_{11}, a_{22}, \ldots$.
If $A$ is square and the __non__diagonal entries are all zero, then
$A$ is called a **diagonal matrix**.
$$
% The Rules create some space below the matrices:
\kern-8ex
\mystack{
\bmat{ccc}
1 & -3/2 & \pi \\
\sqrt{2} & 2.3 & 0
\emat
\Rule{0pt}{0pt}{18pt}
}{\text{not square or diagonal}}
\qquad
\mystack{
\bmat{rr}
1 & 2 \\ 3 & 4
\emat
\Rule{0pt}{0pt}{22pt}
}{\text{square}}
\qquad
\mystack{
\bmat{rr}
1 & 0 \\ 0 & 4
\emat
\Rule{0pt}{0pt}{20pt}
}{\text{diagonal}}
\qquad
\mystack{
\bmat{rr}
1 & 0 \\ 0 & 0
\emat
\Rule{0pt}{0pt}{20pt}
}{\text{diagonal}}
\qquad
\mystack{
\bmat{rr}
2 & 0 \\ 0 & 2
\emat
\Rule{0pt}{0pt}{20pt}
}{\text{scalar}}
$$

**Definition:** A diagonal matrix with all diagonal entries equal is
called a **scalar matrix**. A scalar matrix with diagonal entries all
equal to $1$ is an **identity matrix**.
$$
% The Rules create some space below the matrices:
\mystack{
I_3 = \bmat{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\emat
\Rule{0pt}{0pt}{18pt}
}{\text{identity matrix}}
\qquad
\mystack{
O = \bmat{rrr}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\emat
\Rule{0pt}{0pt}{18pt}
}{\text{scalar}}
$$
Note: Identity $\implies$ scalar $\implies$ diagonal $\implies$ square.

Now we're going to mimick a lot of what we did when we first introduced vectors.

**Definition:** Two matrices are **equal** if they have the same size
and their corresponding entries are equal.
$$
\kern-8ex
% The Rules create some space below the matrices:
\bmat{cc}
1 & -3/2 \\ \sqrt{2} & 0
\emat
\qquad
\bmat{cc}
\cos 0 & -1.5 \\ \sqrt{2} & \sin 0
\emat
\qquad
\bmat{cc}
1 & 2 \\ 3 & 4
\emat
\qquad
\bmat{rrrr}
1 & 2 & 3 & 4
\emat
\qquad
\bmat{r}
1 \\ 2 \\ 3 \\ 4
\emat
$$
The first two above are equal, but no other two are equal.
We distinguish row matrices from column matrices!

**Definition:** If $A$ and $B$ are __both__ $m \times n$ matrices,
then their **sum** $A + B$ is the $m \times n$ matrix obtained by adding the
corresponding entries of $A$ and $B$.
$$
\bmat{rrr}
1 & 2 & 3 \\ 4 & 5 & 6
\emat
+
\bmat{rrr}
0 & -1 & 2 \\ \pi & 0 & -6
\emat
=
\bmat{ccc}
1 & 1 & 5 \\ 4+\pi & 5 & 0
\emat
$$
Using the notation $A = [a_{ij}]$ and $B = [b_{ij}]$, we write
$$
A + B = [a_{ij} + b_{ij}]
\qquad\text{ or }\qquad
(A+B)_{ij} = a_{ij} + b_{ij} .
$$

**Definition:** If $A$ is an $m \times n$ matrix and $c$ is a scalar,
then the **scalar multiple** $cA$ is the $m \times n$ matrix obtained
by multiplying each entry by $c$.
$$
3
\bmat{rrr}
0 & -1 & 2 \\ \pi & 0 & -6
\emat
=
\bmat{rrr}
0 & -3 & 6 \\ 3 \pi & 0 & -18
\emat
$$
We write $cA = [c \, a_{ij}]$ or $(cA)_{ij} = c \, a_{ij}$.

**Definition:** As expected, $-A$ means $(-1)A$ and $A-B$ means $A + (-B)$.

The $m \times n$ **zero matrix** has all entries $0$ and is denoted $O$
or $O_{m\times n}$.
Of course, $A + O = A$.

So we have the real number $0$, the zero vector $\vec 0$ (or $\boldsymbol{0}$ in the text)
and the zero matrix $O$.

**Definition:** If $A$ is $m \times \red{n}$ and $B$ is $\red{n} \times r$, then the **product**
$C = AB$ is the $m \times r$ matrix whose $i,j$ entry is
$$ c_{ij}
= a_{i\red{1}} b_{\red{1}j} + a_{i\red{2}} b_{\red{2}j} + \cdots + a_{i\red{n}} b_{\red{n}j}
= \sum_{\red{k}=1}^{n} a_{i\red{k}} b_{\red{k}j} . $$
This is the dot product of the $i$th row of $A$ with the $j$th column of $B$.

Note that for this to make sense, the number of columns of $A$ must equal the number of rows of $B$. $$ \mystack{A}{m \times n} \ \ \mystack{B}{n \times r} \mystack{=}{\strut} \mystack{AB}{m \times r} $$ This may seem very strange, but it turns out to be useful. We will never use componentwise multiplication, as it is not generally useful.

**Examples on whiteboard:** $2 \times 3$ times $3 \times 4$, $1 \times 3$ times $3 \times 1$,
$3 \times 1$ times $1 \times 3$.

One motivation for this definition of matrix multiplication is that it comes up in linear systems.

**Example 3.8:** Consider the system
$$
\begin{aligned}
4 x + 2 y &= 4 \\
5 x + \ph y &= 8 \\
6 x + 3 y &= 6
\end{aligned}
$$
The left-hand sides are in fact a matrix product:
$$
\bmat{rr}
4 & 2 \\
5 & 1 \\
6 & 3
\emat
\coll x y
$$
Every linear system can be written as $A \vx = \vb$.

**Question:**
If $A$ is an $m \times n$ matrix and $\ve_1$ is the first standard unit vector in $\R^n$,
what is $A \ve_1$?

The answer is an $m \times 1$ column matrix, whose $i$th entry is the dot product
of the $i$th row of $A$ with the vector $\ve_1$.
But $[a_{i1}, a_{i2}, \ldots, a_{in}] \cdot [1, 0, \ldots, 0] = a_{i1}$, the first entry.
So this just "picks out" the first column of $A$.

More generally, we have:

**Theorem 3.1:** If $A$ is $m \times n$, $\ve_i$ is the $i$th $1 \times m$ standard
row vector and $\ve_j$ is the $j$th $n \times 1$ standard column vector, then
$$ \ve_i A = \text{the $i$th row of $A$}
\qquad\text{and}\qquad
A \ve_j = \text{the $j$th column of $A$}.
$$

We write $A^1 = A$ and $A^0 = I_n$.

We will see later that $(AB)C = A(BC)$, so the expression for $A^k$ is unambiguous. And it follows that $$ A^r A^s = A^{r+s} \qquad\text{and}\qquad (A^r)^s = A^{rs} $$ for all nonnegative integers $r$ and $s$.

**Example 3.13 on whiteboard:** Powers of
$$
A = \bmat{rr} 1 & 1 \\ 1 & 1 \emat
$$

.