matrices

Precalculus Notes: Matrices, Matrix Order, Equality, Addition & Subtraction

1. Matrix

A matrix is a rectangular arrangement of numbers enclosed in brackets.

Example

[
A=
\begin{bmatrix}
2&5\
1&4
\end{bmatrix}
]

2. Order (Size) of a Matrix

The order tells how many rows × columns a matrix has.

Always write:

Rows × Columns

Examples

Memory Trick

Rows = Horizontal

Columns = Vertical

Order = Rows first, Columns second

3. Matrix Entries

Each number inside a matrix is called an entry.

Notation

[
a_{ij}
]

where

• (i) = row number

• (j) = column number

Example

[
\begin{bmatrix}
5&9&3\
2&7&8
\end{bmatrix}
]

• (a_{11}=5)

• (a_{12}=9)

• (a_{23}=8)

Row always comes first.

4. Types of Matrices

A. Square Matrix

Rows = Columns

Example

[
\begin{bmatrix}
1&2&3\
4&5&6\
7&8&9
\end{bmatrix}
]

Order:

3 × 3

B. Non-Square Matrix

Rows ≠ Columns

Example

[
\begin{bmatrix}
1&2\
3&4\
5&6
\end{bmatrix}
]

Order:

3 × 2

C. Row Matrix

Exactly one row

Example

[
\begin{bmatrix}
4&7&9
\end{bmatrix}
]

Order:

1 × 3

D. Column Matrix

Exactly one column

Example

[
\begin{bmatrix}
2\
5\
8
\end{bmatrix}
]

Order:

3 × 1

E. Zero Matrix

Every entry equals 0.

Example

[
\begin{bmatrix}
0&0\
0&0
\end{bmatrix}
]

F. Identity Matrix

A square matrix with

• 1's on the diagonal

• 0's everywhere else

Example

[
I_3=
\begin{bmatrix}
1&0&0\
0&1&0\
0&0&1
\end{bmatrix}
]

Identity matrices behave like the number 1 in multiplication.

5. Diagonal Entries

The diagonal runs from

Upper-left

↓

Lower-right

Example

[
\begin{bmatrix}
\boxed1&4&7\
2&\boxed5&9\
6&8&\boxed3
\end{bmatrix}
]

Diagonal entries:

1,5,3

Only square matrices have a full diagonal.

6. Matrix Equality

Two matrices are equal only if BOTH conditions are true.

Condition 1

Same order.

Example

2 × 2 ≠ 2 × 3

Cannot be equal.

Condition 2

Every corresponding entry is equal.

Example

[
\begin{bmatrix}
1&2\
3&4
\end{bmatrix}

\begin{bmatrix}
1&2\
3&4
\end{bmatrix}
]

Equal.

Not equal

[
\begin{bmatrix}
1&2\
3&4
\end{bmatrix}
\neq
\begin{bmatrix}
1&3\
2&4
\end{bmatrix}
]

Same numbers

Wrong positions

Not equal.

7. Matrix Addition

Rule

Matrices can only be added if they have the same order.

Example

[
A=
\begin{bmatrix}
2&5\
3&1
\end{bmatrix}
]

[
B=
\begin{bmatrix}
3&8\
6&2
\end{bmatrix}
]

Add matching entries.

[
A+B=
\begin{bmatrix}
2+3&5+8\
3+6&1+2
\end{bmatrix}

\begin{bmatrix}
5&13\
9&3
\end{bmatrix}
]

Think:

Row with row

Column with column

8. Matrix Subtraction

Rule

Matrices must have the same order.

Subtract corresponding entries.

Example

[
A=
\begin{bmatrix}
1&2\
2&6
\end{bmatrix}
]

[
B=
\begin{bmatrix}
2&-1\
-3&1
\end{bmatrix}
]

[
A-B=
\begin{bmatrix}
1-2&2-(-1)\
2-(-3)&6-1
\end{bmatrix}

\begin{bmatrix}
-1&3\
5&5
\end{bmatrix}
]

Remember

Subtracting a negative

=

Adding

Example

2−(−3)=5

9. Undefined Matrix Addition/Subtraction

Cannot add or subtract matrices of different orders.

Example

2 × 2

3 × 2

Undefined

Example

[
\begin{bmatrix}
1&2\
3&4
\end{bmatrix}
+
\begin{bmatrix}
5\
6
\end{bmatrix}
]

Not possible.

10. Properties of Matrix Addition

Commutative

Order does not matter.

[
A+B=B+A
]

Associative

Grouping does not matter.

[
(A+B)+C=A+(B+C)
]

Subtraction is NOT commutative.

Usually

[
A-B\neq B-A
]

11. Summary Table

12. Important Formulas

Matrix Addition

[
A+B=
[a_{ij}+b_{ij}]
]

Matrix Subtraction

[
A-B=
[a_{ij}-b_{ij}]
]

13. Exam Tips

✓ Order = Rows × Columns.

✓ Rows come before columns.

✓ (a_{ij}):

• (i) = row

• (j) = column

✓ Square matrix → rows = columns.

✓ Identity matrix → diagonal = 1, everything else = 0.

✓ Zero matrix → every entry = 0.

✓ Matrices must have the same order to add or subtract.

✓ Add and subtract entry by entry.

✓ Matrix addition is commutative and associative.

✓ Matrix subtraction is not commutative.

✓ Two matrices are equal only if:

1. Same order.

2. Every corresponding entry is equal.