Introduction

Matrix multiplication is a way to combine two transformations into a single transformation.
If you already know how a matrix acts on a vector—turning $v$ into $Av$—then matrix multiplication tells you how to apply two such actions in a row using one new matrix.

This idea is powerful because it lets us:

• Chain multiple transformations together
• Simplify repeated operations
• Understand how geometry changes under sequences of moves

Why Combine Transformations?

Some common reasons:

• Applying a rotation then a scaling
• Applying a shear then a reflection
• Applying a camera transform then a projection (in graphics)
• Applying a coordinate change then a measurement

Instead of computing each step separately, we can build one matrix that does everything at once.

Matrix Multiplication as “Do This, Then That”

Suppose:

• $A$ is a matrix that transforms vectors first
• $B$ is a matrix that transforms vectors second

If you apply $A$ then $B$ to a vector $v$, you compute: $$B(Av)$$ Matrix multiplication defines a new matrix $C$ such that: $$C v = B(Av)$$ So the combined matrix is: $$C = BA$$ Important:
The order is right-to-left because the vector is on the right.

How Matrix Multiplication Works

To multiply two matrices:

• Take each row of the left matrix
• Take each column of the right matrix
• Multiply matching entries
• Add the results

For example, if $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad B = \begin{pmatrix} p & q \\ r & s \end{pmatrix}$$ Then: $$BA = \begin{pmatrix} pa + qc & pb + qd \\ ra + sc & rb + sd \end{pmatrix}$$ You don’t need to memorize this—just remember:

• Rows of the first matrix
• Columns of the second matrix
• Multiply + add

Geometric Meaning

Matrix multiplication corresponds to doing one geometric action after another.

Examples:

• Rotate then scale
• Shear then rotate
• Reflect then shear

Each combined action is still linear, so it can be represented by a single matrix.

Common Mistakes

• Order matters.
  In general, $AB \neq BA$.
• You can only multiply if the inner dimensions match.
  If $A$ is $m \times n$ and $B$ is $n \times p$, then $BA$ is defined.
• Matrix multiplication is not element‑wise.
  It’s row–column multiplication.

Calculator

Exercises

1. Multiply the matrices $$\begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix} \begin{pmatrix}3 & 0 \\ 1 & 4\end{pmatrix}$$

   Solution

   $$\begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix} \begin{pmatrix}3 & 0 \\ 1 & 4\end{pmatrix} = \begin{pmatrix} 1\cdot 3 + 2\cdot 1 & 1\cdot 0 + 2\cdot 4 \\ 0\cdot 3 + 1\cdot 1 & 0\cdot 0 + 1\cdot 4 \end{pmatrix} = \begin{pmatrix}5 & 8 \\ 1 & 4\end{pmatrix}$$
2. Compute the product $$\begin{pmatrix}2 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}4 \\ 3\end{pmatrix}$$

   Solution

   $$\begin{pmatrix}2 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}4 \\ 3\end{pmatrix} = \begin{pmatrix}2\cdot 4 + 1\cdot 3 \\ 1\cdot 4 + 0\cdot 3\end{pmatrix} = \begin{pmatrix}11 \\ 4\end{pmatrix}$$
3. Let $A$ be a rotation matrix and $B$ be a scaling matrix.
   Explain in words what the matrix $BA$ represents.

   Solution

   
   $BA$ means “rotate, then scale.”
   The combined matrix performs both actions in that order.
4. Determine whether the product $AB$ is defined if
   $A$ is $2 \times 3$ and $B$ is $3 \times 2$.

   Solution

   
   Yes, $AB$ is defined because the inner dimensions match ($3=3$).
   The result is a $2 \times 2$ matrix.
5. tranform the vector by the matrix $$\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}5 \\ 7\end{pmatrix}$$

   Solution

   $$\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix}5 \\ 7\end{pmatrix} = \begin{pmatrix}7 \\ 5\end{pmatrix}$$ This matrix swaps the two entries.
6. True or false: If $AB$ is defined, then $BA$ is always defined.

   Solution

   
   False.
   Even if $AB$ is defined, $BA$ may not be.
7. Compute $$\begin{pmatrix}3 & -1 \\ 2 & 4\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & 2\end{pmatrix}$$

   Solution

   $$\begin{pmatrix}3 & -1 \\ 2 & 4\end{pmatrix} \begin{pmatrix}1 & 0 \\ 0 & 2\end{pmatrix} = \begin{pmatrix} 3\cdot 1 + (-1)\cdot 0 & 3\cdot 0 + (-1)\cdot 2 \\ 2\cdot 1 + 4\cdot 0 & 2\cdot 0 + 4\cdot 2 \end{pmatrix} = \begin{pmatrix}3 & -2 \\ 2 & 8\end{pmatrix}$$