Matrix Basics and Operations

1. Matrices and Matrix Operations: The Basics

Now we formally define the tool we've been using!

Definition of a Matrix

A matrix is a rectangular array of numbers. The numbers inside the matrix are called the entries.

Size of a Matrix

The size of a matrix is always described as:

(Hint: Rows run horizontally $\leftrightarrow$, Columns run vertically $\updownarrow$)

An $m \times n$ matrix is a matrix having $m$ rows and $n$ columns.

2. Matrix Terminology: Square Matrices and the Trace

Square Matrix

A matrix is a square matrix if the number of rows equals the number of columns (e.g., $2 \times 2$, $3 \times 3$).

Zero Matrix

A matrix where every single entry is $0$. Denoted as $\mathbf{0}_{m \times n}$.

The Main Diagonal

In a square matrix, the main diagonal is the line of entries starting from the top-left corner down to the bottom-right corner. (Entries $a_{11}, a_{22}, a_{33} \dots a_{nn}$).

Remark: A non-square matrix does NOT have a main diagonal.

The Trace of a Matrix ($\text{tr}(A)$)

Let $A$ be a square matrix. The trace of $A$ is defined as the sum of all the entries on its main diagonal:

Remark: The trace of a non-square matrix is undefined.

3. Matrix Arithmetic: Addition, Subtraction, and Scalars

Addition ($A + B$) and Subtraction ($A - B$)

To add or subtract two matrices, you simply add or subtract the corresponding entries in the exact same positions.

Crucial Rule: Matrices must be the exact same size to be added or subtracted.

Scalar Multiplication ($cA$)

Let $c$ be a scalar (a regular real number). To find $cA$, you multiply every single entry inside matrix $A$ by the number $c$.

4. Matrix Multiplication ($A \cdot B$)

Matrix multiplication is highly specific and does not work like normal multiplication.

The Dimension Rule

Let $A$ be an $(m \times r)$ matrix and $B$ be an $(r \times n)$ matrix. You can only multiply them if the middle two numbers match (the columns of $A$ must equal the rows of $B$).

The result, $AB$, will be a new matrix with the outer dimensions: $(m \times n)$.

How to Multiply ("Row by Column")

To find the entry in the $i$-th row and $j$-th column of your new matrix $AB$:

1. Single out the $i$-th row from matrix $A$.
2. Single out the $j$-th column from matrix $B$.
3. Multiply their corresponding entries together one by one, and add the resulting products.

Important Note: Unlike regular numbers where $3 \times 4 = 4 \times 3$, in matrices, $AB \neq BA$! Sometimes $AB$ works, but if you flip them, the dimensions no longer match and $BA$ is "undefined".