Topic 4.10 Notes – Matrices

What Matrices Are

A matrix is a rectangular array of values arranged in rows and columns. If a matrix has 2 rows and 3 columns, its dimensions are $2\times 3$. Rows come first.

$$A=\begin{bmatrix} 4 & -1 & 6\\ 2 & 0 & 3 \end{bmatrix}$$

Here, $A$ is $2\times 3$. An entry is named by position. In $a_{ij}$, $i$ is the row and $j$ is the column, so $a_{12}=-1$ and $a_{23}=3$.

Matrix entries by row and column position

The diagram shows the indexing convention you'll use throughout this unit. $a_{ij}$ means row $i$, column $j$.

Quick recognition types:

• Row matrix has one row, like $\begin{bmatrix} 2 & 5 & -1 \end{bmatrix}$
• Column matrix has one column, like $\begin{bmatrix} 3 \\ -4 \\ 7 \end{bmatrix}$
• Square matrix has the same number of rows and columns, like $3\times 3$

A column can also represent a vector. That matters because matrix multiplication uses dot products, which means multiplying matching entries and adding.

When a Matrix Product Is Defined

For matrices with sizes $n\times m$ and $m\times p$, the product is defined:

$$(n\times m)(m\times p)\to n\times p$$

Two things to remember:

• Inner dimensions must match. Those are the middle numbers.
• Outer dimensions give the size of the product.

Why? Each entry in the product comes from one row of the first matrix and one column of the second. Those must have the same number of entries to form a dot product.

Examples:

A common mistake is checking the outer dimensions instead of the inner ones.

How to Multiply Matrices

The rule is row by column. If $C=AB$, then entry $c_{ij}$ comes from row $i$ of $A$ and column $j$ of $B$.

$$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{im}b_{mj}$$

That formula just says “take a dot product.”

Worked example

$$A=\begin{bmatrix} 2 & -1 & 4\\ 0 & 3 & 5 \end{bmatrix}, \qquad B=\begin{bmatrix} 1 & 2\\ -2 & 0\\ 3 & -1 \end{bmatrix}$$

$A$ is $2\times 3$, $B$ is $3\times 2$, so $AB$ is $2\times 2$.

Here are the four dot products that fill the entries of $AB$.

$$c_{11}=2(1)+(-1)(-2)+4(3)=16$$ $$c_{12}=2(2)+(-1)(0)+4(-1)=0$$ $$c_{21}=0(1)+3(-2)+5(3)=9$$ $$c_{22}=0(2)+3(0)+5(-1)=-5$$

So

$$AB=\begin{bmatrix} 16 & 0\\ 9 & -5 \end{bmatrix}$$

The location matters. Row 2 with column 1 goes in entry $(2,1)$. This is not entry-by-entry multiplication.

Order Matters in Matrix Multiplication

With matrices $A$ and $B$, three things can happen:

• both $AB$ and $BA$ exist
• one exists and the other does not
• both are undefined

In general,

$$AB\ne BA$$

Example with both defined:

$$A=\begin{bmatrix}1&2\\0&1\end{bmatrix},\quad B=\begin{bmatrix}2&0\\3&1\end{bmatrix}$$

Then $AB=\begin{bmatrix}8&2\\3&1\end{bmatrix}$ but $BA=\begin{bmatrix}2&4\\3&7\end{bmatrix}$. Same sizes, different answers.

If $A$ is $2\times 3$ and $B$ is $3\times 4$, then $AB$ is defined but $BA$ is undefined.

On a quiz or FRQ, keep the order exactly as written.

Interpreting Matrix Products and Avoiding Mistakes

A matrix product often means “combine categories into totals.”

If

$$Q=\begin{bmatrix} 12 & 8 & 5\\ 7 & 10 & 9 \end{bmatrix}$$

is quantities sold at two locations for three products, and

$$p=\begin{bmatrix} 3\\ 2\\ 4 \end{bmatrix}$$

is the price column, then

$$Qp=\begin{bmatrix} 72\\ 77 \end{bmatrix}$$

That $2\times 1$ result means one total revenue for each of the two locations.

Technology follows the same rules. A dimension error means the product is undefined.

Common mistakes:

• writing dimensions backward
• checking outer instead of inner dimensions
• giving the wrong product size
• multiplying entry by entry
• putting a correct dot product in the wrong spot
• assuming $AB=BA$ or that $BA$ must exist if $AB$ does