matrix multiplication
Precalculus Notes: Matrix Multiplication
1. What is Matrix Multiplication?
Matrix multiplication combines two matrices to produce a new matrix.
Unlike scalar multiplication, you do NOT multiply every entry directly.
Instead:
Multiply rows of the first matrix by columns of the second matrix.
Add the products.
2. When is Matrix Multiplication Possible?
Suppose
[
A_{m\times n}
]
and
[
B_{p\times q}
]
The product
[
AB
]
is defined only if
[
\boxed{n=p}
]
Meaning:
Columns of the first matrix = Rows of the second matrix
Easy Memory Trick
Inside numbers must match.
Example
[
(2\times3)(3\times4)
]
Inside numbers
[
3=3
]
Multiplication is possible.
Result
[
2\times4
]
Example
[
(2\times3)(4\times2)
]
Inside numbers
[
3\neq4
]
Cannot multiply.
3. Size of the Product Matrix
After multiplication,
keep the outside numbers.
Example
[
(4\times2)(2\times5)
]
Result
[
4\times5
]
Summary
Matrix A | Matrix B | Can Multiply? | Result |
|---|---|---|---|
2×3 | 3×4 | Yes | 2×4 |
3×5 | 5×2 | Yes | 3×2 |
2×2 | 2×2 | Yes | 2×2 |
2×3 | 4×2 | No | Undefined |
4. Rule for Matrix Multiplication
Each entry comes from
Row × Column
Multiply corresponding entries.
Then add.
Formula
[
c_{ij}
a_{i1}b_{1j}
+
a_{i2}b_{2j}
+\cdots
]
Think
Row • Column
(dot product)
5. Example: Row × Column
[
A=
\begin{bmatrix}
1&3
\end{bmatrix}
]
[
B=
\begin{bmatrix}
2\
4
\end{bmatrix}
]
Multiply
[
AB
]
Step 1
Multiply corresponding entries
[
1(2)+3(4)
]
Step 2
Compute
[
2+12=14
]
Answer
[
\boxed{
\begin{bmatrix}
14
\end{bmatrix}}
]
6. Multiplying Two 2×2 Matrices
Example
[
A=
\begin{bmatrix}
0&1\
2&3
\end{bmatrix}
]
[
B=
\begin{bmatrix}
4&5\
6&7
\end{bmatrix}
]
Find Entry (1,1)
Row 1 × Column 1
[
0(4)+1(6)=6
]
Find Entry (1,2)
Row 1 × Column 2
[
0(5)+1(7)=7
]
Find Entry (2,1)
Row 2 × Column 1
[
2(4)+3(6)
]
[
8+18=26
]
Find Entry (2,2)
Row 2 × Column 2
[
2(5)+3(7)
]
[
10+21=31
]
Final Answer
[
AB=
\begin{bmatrix}
6&7\
26&31
\end{bmatrix}
]
7. Step-by-Step Algorithm
Suppose
[
A=
\begin{bmatrix}
a&b\
c&d
\end{bmatrix}
]
[
B=
\begin{bmatrix}
e&f\
g&h
\end{bmatrix}
]
Step 1
Top-left
Row 1 × Column 1
[
ae+bg
]
Step 2
Top-right
Row 1 × Column 2
[
af+bh
]
Step 3
Bottom-left
Row 2 × Column 1
[
ce+dg
]
Step 4
Bottom-right
Row 2 × Column 2
[
cf+dh
]
8. Matrix Multiplication is NOT Commutative
For real numbers
[
ab=ba
]
For matrices
[
\boxed{AB\neq BA}
]
Usually
they give different answers.
Sometimes
one exists while the other doesn't.
Example
[
A=
\begin{bmatrix}
2&1\
3&2
\end{bmatrix}
]
[
B=
\begin{bmatrix}
1&0\
4&3
\end{bmatrix}
]
[
AB=
\begin{bmatrix}
6&3\
11&6
\end{bmatrix}
]
But
[
BA=
\begin{bmatrix}
2&1\
17&10
\end{bmatrix}
]
Since
[
AB\neq BA
]
Matrix multiplication is not commutative.
9. Example of Different Sizes
[
A=
\begin{bmatrix}
3&2&8\
4&1&6
\end{bmatrix}
]
Order
[
2\times3
]
[
B=
\begin{bmatrix}
2&4\
3&1\
2&3
\end{bmatrix}
]
Order
[
3\times2
]
Can multiply?
Yes
because
[
3=3
]
Result size
[
2\times2
]
Entry (1,1)
[
3(2)+2(3)+8(2)
]
[
6+6+16=28
]
Entry (1,2)
[
3(4)+2(1)+8(3)
]
[
12+2+24=38
]
Entry (2,1)
[
4(2)+1(3)+6(2)
]
[
8+3+12=23
]
Entry (2,2)
[
4(4)+1(1)+6(3)
]
[
16+1+18=35
]
Answer
[
AB=
\begin{bmatrix}
28&38\
23&35
\end{bmatrix}
]
10. Applications
Matrix multiplication is used in
Economics
Finance
Engineering
Computer graphics
Artificial Intelligence
Data science
Physics
Robotics
Cryptography
Population models
11. Restaurant Revenue Example
Matrix A
Prices
[
2\times3
]
Matrix B
Items sold
[
3\times2
]
Multiply
[
AB
]
Result
[
2\times2
]
Rows
Restaurants
Columns
Days
Example result
[
\begin{bmatrix}
28&38\
23&35
\end{bmatrix}
]
Means
Restaurant | Monday | Tuesday |
|---|---|---|
Y | $28,000 | $38,000 |
Z | $23,000 | $35,000 |
12. Vehicle Survey Example
Matrix A
People by
Gender
Age
Matrix B
Vehicle preference percentages
Multiply
[
AB
]
Result
Expected buyers by
Gender
Vehicle type
13. Cryptography
Matrices can encrypt messages.
Steps
Assign numbers to letters.
Example
A = 1
B = 2
...
Z = 26
Space = 0
Create a square encryption matrix.
Example
[
\begin{bmatrix}
1&2\
2&1
\end{bmatrix}
]
Convert letters into numbers.
Arrange numbers into matrices.
Multiply by the secret matrix.
Encrypted message produced.
Receiver multiplies by the inverse matrix.
Original message returns.
14. Common Mistakes
Mistake 1
Trying to multiply incompatible matrices.
Always check
Columns of first = Rows of second.
Mistake 2
Multiplying entry-by-entry.
Wrong.
Always use
Row × Column.
Mistake 3
Wrong size of answer.
Keep
Outside numbers.
Example
[
(2\times4)(4\times6)
]
Answer is
[
2\times6
]
Mistake 4
Thinking
[
AB=BA
]
Usually false.
15. Quick Reference
Matrix Multiplication Rule
Columns of first = Rows of second
Product Size
Keep outside numbers.
Entry Formula
[
\boxed{\text{Row}\times\text{Column}}
]
Multiply corresponding entries.
Add.
Commutative?
[
\boxed{AB\neq BA}
]
Generally false.
Formula Sheet
Condition
[
A_{m\times n}B_{p\times q}
]
Possible only if
[
n=p
]
Product Size
[
(m\times n)(n\times p)
m\times p
]
Entry Formula
[
c_{ij}
a_{i1}b_{1j}
+
a_{i2}b_{2j}
+\cdots
+
a_{in}b_{nj}
]
Memory Tricks
Inside numbers match → Multiplication is possible.
Outside numbers stay → Product size.
Row × Column → Every entry.
Multiply, then add → Dot product.
AB ≠ BA → Matrix multiplication is not commutative.