Matrices - Ninth Grade Math
Introduction to Matrices
Matrix: A rectangular array of numbers arranged in rows and columns
Plural: Matrices
Notation: Usually represented by capital letters (A, B, C, etc.)
Uses: Organizing data, solving systems of equations, transformations, computer graphics
Plural: Matrices
Notation: Usually represented by capital letters (A, B, C, etc.)
Uses: Organizing data, solving systems of equations, transformations, computer graphics
1. Matrix Vocabulary
Basic Matrix Structure
General Matrix Notation:
$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \end{bmatrix}$$
Element Notation: $a_{ij}$
• $i$ = row number (first subscript)
• $j$ = column number (second subscript)
• $a_{ij}$ = element in row $i$, column $j$
$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33} & \cdots & a_{3n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & a_{m3} & \cdots & a_{mn} \end{bmatrix}$$
Element Notation: $a_{ij}$
• $i$ = row number (first subscript)
• $j$ = column number (second subscript)
• $a_{ij}$ = element in row $i$, column $j$
Dimensions (Order/Size)
Dimensions: Size of a matrix written as $m \times n$ (read "m by n")
m: Number of rows
n: Number of columns
Order: Always rows first, then columns
Example: A $3 \times 4$ matrix has 3 rows and 4 columns
m: Number of rows
n: Number of columns
Order: Always rows first, then columns
Example: A $3 \times 4$ matrix has 3 rows and 4 columns
Example 1: Identify dimensions and elements
$$A = \begin{bmatrix} 2 & 5 & -1 \\ 0 & 3 & 7 \end{bmatrix}$$
Dimensions: $2 \times 3$ (2 rows, 3 columns)
Element $a_{12}$: Row 1, Column 2 = 5
Element $a_{23}$: Row 2, Column 3 = 7
Element $a_{11}$: Row 1, Column 1 = 2
$$A = \begin{bmatrix} 2 & 5 & -1 \\ 0 & 3 & 7 \end{bmatrix}$$
Dimensions: $2 \times 3$ (2 rows, 3 columns)
Element $a_{12}$: Row 1, Column 2 = 5
Element $a_{23}$: Row 2, Column 3 = 7
Element $a_{11}$: Row 1, Column 1 = 2
Special Types of Matrices
Square Matrix: Number of rows = number of columns ($n \times n$)
Example: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is $2 \times 2$
Row Matrix: Matrix with only 1 row ($1 \times n$)
Example: $\begin{bmatrix} 5 & 2 & 8 \end{bmatrix}$ is $1 \times 3$
Column Matrix: Matrix with only 1 column ($m \times 1$)
Example: $\begin{bmatrix} 3 \\ 7 \\ 1 \end{bmatrix}$ is $3 \times 1$
Zero Matrix (Null Matrix): All elements are zero
Example: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
Identity Matrix ($I_n$): Square matrix with 1's on main diagonal, 0's elsewhere
Example: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Diagonal Matrix: Square matrix with non-zero elements only on main diagonal
Example: $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix}$
Example: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is $2 \times 2$
Row Matrix: Matrix with only 1 row ($1 \times n$)
Example: $\begin{bmatrix} 5 & 2 & 8 \end{bmatrix}$ is $1 \times 3$
Column Matrix: Matrix with only 1 column ($m \times 1$)
Example: $\begin{bmatrix} 3 \\ 7 \\ 1 \end{bmatrix}$ is $3 \times 1$
Zero Matrix (Null Matrix): All elements are zero
Example: $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
Identity Matrix ($I_n$): Square matrix with 1's on main diagonal, 0's elsewhere
Example: $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
Diagonal Matrix: Square matrix with non-zero elements only on main diagonal
Example: $\begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix}$
Main Diagonal: Elements where row number = column number ($a_{11}, a_{22}, a_{33}$, etc.)
Scalar: A single number (not a matrix)
Equal Matrices: Two matrices with same dimensions and identical corresponding elements
Scalar: A single number (not a matrix)
Equal Matrices: Two matrices with same dimensions and identical corresponding elements
2. Matrix Operation Rules
Rule 1: Addition/Subtraction
• Matrices must have the SAME DIMENSIONS
• Add/subtract corresponding elements
• If dimensions differ, operation is UNDEFINED
Rule 2: Scalar Multiplication
• Can multiply any matrix by a scalar
• Multiply every element by the scalar
• No dimension restrictions
Rule 3: Matrix Multiplication
• Number of COLUMNS in first matrix must equal number of ROWS in second
• If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$
• Order matters: $AB \neq BA$ (usually)
• If dimensions don't match properly, multiplication is UNDEFINED
• Matrices must have the SAME DIMENSIONS
• Add/subtract corresponding elements
• If dimensions differ, operation is UNDEFINED
Rule 2: Scalar Multiplication
• Can multiply any matrix by a scalar
• Multiply every element by the scalar
• No dimension restrictions
Rule 3: Matrix Multiplication
• Number of COLUMNS in first matrix must equal number of ROWS in second
• If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$
• Order matters: $AB \neq BA$ (usually)
• If dimensions don't match properly, multiplication is UNDEFINED
Compatibility Check:
For $A_{m \times n} \times B_{p \times q}$:
• Must have: $n = p$ (middle numbers match)
• Result: $m \times q$ (outer numbers)
For $A_{m \times n} \times B_{p \times q}$:
• Must have: $n = p$ (middle numbers match)
• Result: $m \times q$ (outer numbers)
3. Add and Subtract Matrices
Matrix Addition Formula:
If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$ matrices, then:
$$A + B = [a_{ij} + b_{ij}]$$
In words: Add corresponding elements
Matrix Subtraction Formula:
$$A - B = [a_{ij} - b_{ij}]$$
In words: Subtract corresponding elements
If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$ matrices, then:
$$A + B = [a_{ij} + b_{ij}]$$
In words: Add corresponding elements
Matrix Subtraction Formula:
$$A - B = [a_{ij} - b_{ij}]$$
In words: Subtract corresponding elements
Steps to Add/Subtract Matrices:
Step 1: Check that matrices have same dimensions
Step 2: Add/subtract elements in same position
Step 3: Place result in same position in answer matrix
Step 4: Continue for all elements
Step 1: Check that matrices have same dimensions
Step 2: Add/subtract elements in same position
Step 3: Place result in same position in answer matrix
Step 4: Continue for all elements
Example 1: Matrix Addition
$$A = \begin{bmatrix} 2 & 5 \\ 3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 2 \\ 0 & 6 \end{bmatrix}$$
Find $A + B$:
$$A + B = \begin{bmatrix} 2+4 & 5+2 \\ 3+0 & 1+6 \end{bmatrix} = \begin{bmatrix} 6 & 7 \\ 3 & 7 \end{bmatrix}$$
$$A = \begin{bmatrix} 2 & 5 \\ 3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 2 \\ 0 & 6 \end{bmatrix}$$
Find $A + B$:
$$A + B = \begin{bmatrix} 2+4 & 5+2 \\ 3+0 & 1+6 \end{bmatrix} = \begin{bmatrix} 6 & 7 \\ 3 & 7 \end{bmatrix}$$
Example 2: Matrix Subtraction
$$C = \begin{bmatrix} 8 & 3 & 5 \\ 1 & 4 & 2 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & 1 & 3 \\ 5 & 0 & 4 \end{bmatrix}$$
Find $C - D$:
$$C - D = \begin{bmatrix} 8-2 & 3-1 & 5-3 \\ 1-5 & 4-0 & 2-4 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 2 \\ -4 & 4 & -2 \end{bmatrix}$$
$$C = \begin{bmatrix} 8 & 3 & 5 \\ 1 & 4 & 2 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & 1 & 3 \\ 5 & 0 & 4 \end{bmatrix}$$
Find $C - D$:
$$C - D = \begin{bmatrix} 8-2 & 3-1 & 5-3 \\ 1-5 & 4-0 & 2-4 \end{bmatrix} = \begin{bmatrix} 6 & 2 & 2 \\ -4 & 4 & -2 \end{bmatrix}$$
Example 3: Undefined operation
$$E = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}_{2 \times 2}, \quad F = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}_{2 \times 3}$$
Can we find $E + F$?
NO! Different dimensions: $2 \times 2 \neq 2 \times 3$
Answer: $E + F$ is UNDEFINED
$$E = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}_{2 \times 2}, \quad F = \begin{bmatrix} 5 & 6 & 7 \\ 8 & 9 & 10 \end{bmatrix}_{2 \times 3}$$
Can we find $E + F$?
NO! Different dimensions: $2 \times 2 \neq 2 \times 3$
Answer: $E + F$ is UNDEFINED
4. Multiply a Matrix by a Scalar
Scalar Multiplication: Multiplying every element of a matrix by a single number (scalar)
Scalar Multiplication Formula:
If $k$ is a scalar and $A = [a_{ij}]$ is an $m \times n$ matrix, then:
$$kA = [k \cdot a_{ij}]$$
In words: Multiply every element by the scalar
Result: Same dimensions as original matrix
If $k$ is a scalar and $A = [a_{ij}]$ is an $m \times n$ matrix, then:
$$kA = [k \cdot a_{ij}]$$
In words: Multiply every element by the scalar
Result: Same dimensions as original matrix
Example 1: Scalar multiplication
$$A = \begin{bmatrix} 2 & -3 & 5 \\ 0 & 4 & 1 \end{bmatrix}, \quad k = 3$$
Find $3A$:
$$3A = \begin{bmatrix} 3(2) & 3(-3) & 3(5) \\ 3(0) & 3(4) & 3(1) \end{bmatrix} = \begin{bmatrix} 6 & -9 & 15 \\ 0 & 12 & 3 \end{bmatrix}$$
$$A = \begin{bmatrix} 2 & -3 & 5 \\ 0 & 4 & 1 \end{bmatrix}, \quad k = 3$$
Find $3A$:
$$3A = \begin{bmatrix} 3(2) & 3(-3) & 3(5) \\ 3(0) & 3(4) & 3(1) \end{bmatrix} = \begin{bmatrix} 6 & -9 & 15 \\ 0 & 12 & 3 \end{bmatrix}$$
Example 2: Negative scalar
$$B = \begin{bmatrix} 4 & 1 \\ -2 & 5 \end{bmatrix}$$
Find $-2B$:
$$-2B = \begin{bmatrix} -2(4) & -2(1) \\ -2(-2) & -2(5) \end{bmatrix} = \begin{bmatrix} -8 & -2 \\ 4 & -10 \end{bmatrix}$$
$$B = \begin{bmatrix} 4 & 1 \\ -2 & 5 \end{bmatrix}$$
Find $-2B$:
$$-2B = \begin{bmatrix} -2(4) & -2(1) \\ -2(-2) & -2(5) \end{bmatrix} = \begin{bmatrix} -8 & -2 \\ 4 & -10 \end{bmatrix}$$
Example 3: Fraction scalar
$$C = \begin{bmatrix} 6 & 9 \\ 12 & 3 \end{bmatrix}$$
Find $\frac{1}{3}C$:
$$\frac{1}{3}C = \begin{bmatrix} \frac{1}{3}(6) & \frac{1}{3}(9) \\ \frac{1}{3}(12) & \frac{1}{3}(3) \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}$$
$$C = \begin{bmatrix} 6 & 9 \\ 12 & 3 \end{bmatrix}$$
Find $\frac{1}{3}C$:
$$\frac{1}{3}C = \begin{bmatrix} \frac{1}{3}(6) & \frac{1}{3}(9) \\ \frac{1}{3}(12) & \frac{1}{3}(3) \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix}$$
5. Add and Subtract Scalar Multiples of Matrices
Linear Combination: Expression involving scalar multiplication and addition/subtraction of matrices
Form: $kA + lB$ or $kA - lB$ where $k$ and $l$ are scalars
Form: $kA + lB$ or $kA - lB$ where $k$ and $l$ are scalars
Steps for Scalar Multiples:
Step 1: Perform scalar multiplication first
Step 2: Then add or subtract the resulting matrices
Step 3: Simplify
Order of Operations: Scalar multiplication before addition/subtraction
Step 1: Perform scalar multiplication first
Step 2: Then add or subtract the resulting matrices
Step 3: Simplify
Order of Operations: Scalar multiplication before addition/subtraction
Example 1: Find $2A + 3B$
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 0 \\ 1 & 2 \end{bmatrix}$$
Step 1: Find $2A$
$$2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$$
Step 2: Find $3B$
$$3B = \begin{bmatrix} 15 & 0 \\ 3 & 6 \end{bmatrix}$$
Step 3: Add
$$2A + 3B = \begin{bmatrix} 2+15 & 4+0 \\ 6+3 & 8+6 \end{bmatrix} = \begin{bmatrix} 17 & 4 \\ 9 & 14 \end{bmatrix}$$
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 0 \\ 1 & 2 \end{bmatrix}$$
Step 1: Find $2A$
$$2A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$$
Step 2: Find $3B$
$$3B = \begin{bmatrix} 15 & 0 \\ 3 & 6 \end{bmatrix}$$
Step 3: Add
$$2A + 3B = \begin{bmatrix} 2+15 & 4+0 \\ 6+3 & 8+6 \end{bmatrix} = \begin{bmatrix} 17 & 4 \\ 9 & 14 \end{bmatrix}$$
Example 2: Find $4C - 2D$
$$C = \begin{bmatrix} 3 & 1 \\ 0 & 5 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & 4 \\ 6 & 1 \end{bmatrix}$$
$$4C = \begin{bmatrix} 12 & 4 \\ 0 & 20 \end{bmatrix}, \quad 2D = \begin{bmatrix} 4 & 8 \\ 12 & 2 \end{bmatrix}$$
$$4C - 2D = \begin{bmatrix} 12-4 & 4-8 \\ 0-12 & 20-2 \end{bmatrix} = \begin{bmatrix} 8 & -4 \\ -12 & 18 \end{bmatrix}$$
$$C = \begin{bmatrix} 3 & 1 \\ 0 & 5 \end{bmatrix}, \quad D = \begin{bmatrix} 2 & 4 \\ 6 & 1 \end{bmatrix}$$
$$4C = \begin{bmatrix} 12 & 4 \\ 0 & 20 \end{bmatrix}, \quad 2D = \begin{bmatrix} 4 & 8 \\ 12 & 2 \end{bmatrix}$$
$$4C - 2D = \begin{bmatrix} 12-4 & 4-8 \\ 0-12 & 20-2 \end{bmatrix} = \begin{bmatrix} 8 & -4 \\ -12 & 18 \end{bmatrix}$$
6. Multiply Two Matrices
Matrix Multiplication: Combining two matrices to produce a third matrix
Important: NOT done element-by-element!
Method: Row × Column multiplication
Important: NOT done element-by-element!
Method: Row × Column multiplication
Matrix Multiplication Formula:
If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$
Element formula:
$$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$$
In words: Element in row $i$, column $j$ of $AB$ equals:
(Row $i$ of $A$) · (Column $j$ of $B$)
Process: Multiply corresponding elements and add
If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$
Element formula:
$$(AB)_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$$
In words: Element in row $i$, column $j$ of $AB$ equals:
(Row $i$ of $A$) · (Column $j$ of $B$)
Process: Multiply corresponding elements and add
Steps to Multiply Matrices:
Step 1: Check compatibility (columns of A = rows of B)
Step 2: Determine dimensions of result
Step 3: For each element in result:
a) Take corresponding row from first matrix
b) Take corresponding column from second matrix
c) Multiply pairs and add
Step 4: Place result in correct position
Step 1: Check compatibility (columns of A = rows of B)
Step 2: Determine dimensions of result
Step 3: For each element in result:
a) Take corresponding row from first matrix
b) Take corresponding column from second matrix
c) Multiply pairs and add
Step 4: Place result in correct position
Example 1: Basic matrix multiplication
$$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}_{2 \times 2}, \quad B = \begin{bmatrix} 5 & 1 \\ 0 & 2 \end{bmatrix}_{2 \times 2}$$
Check: $2 \times 2$ and $2 \times 2$ → result is $2 \times 2$ ✓
Calculate each element:
$(AB)_{11}$: Row 1 of A × Column 1 of B
$= (2)(5) + (3)(0) = 10 + 0 = 10$
$(AB)_{12}$: Row 1 of A × Column 2 of B
$= (2)(1) + (3)(2) = 2 + 6 = 8$
$(AB)_{21}$: Row 2 of A × Column 1 of B
$= (1)(5) + (4)(0) = 5 + 0 = 5$
$(AB)_{22}$: Row 2 of A × Column 2 of B
$= (1)(1) + (4)(2) = 1 + 8 = 9$
$$AB = \begin{bmatrix} 10 & 8 \\ 5 & 9 \end{bmatrix}$$
$$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}_{2 \times 2}, \quad B = \begin{bmatrix} 5 & 1 \\ 0 & 2 \end{bmatrix}_{2 \times 2}$$
Check: $2 \times 2$ and $2 \times 2$ → result is $2 \times 2$ ✓
Calculate each element:
$(AB)_{11}$: Row 1 of A × Column 1 of B
$= (2)(5) + (3)(0) = 10 + 0 = 10$
$(AB)_{12}$: Row 1 of A × Column 2 of B
$= (2)(1) + (3)(2) = 2 + 6 = 8$
$(AB)_{21}$: Row 2 of A × Column 1 of B
$= (1)(5) + (4)(0) = 5 + 0 = 5$
$(AB)_{22}$: Row 2 of A × Column 2 of B
$= (1)(1) + (4)(2) = 1 + 8 = 9$
$$AB = \begin{bmatrix} 10 & 8 \\ 5 & 9 \end{bmatrix}$$
Example 2: Non-square matrices
$$C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2 \times 3}, \quad D = \begin{bmatrix} 2 & 0 \\ 1 & 3 \\ 4 & 1 \end{bmatrix}_{3 \times 2}$$
Check: $2 \times 3$ and $3 \times 2$ → result is $2 \times 2$ ✓
$$(CD)_{11} = (1)(2) + (2)(1) + (3)(4) = 2 + 2 + 12 = 16$$
$$(CD)_{12} = (1)(0) + (2)(3) + (3)(1) = 0 + 6 + 3 = 9$$
$$(CD)_{21} = (4)(2) + (5)(1) + (6)(4) = 8 + 5 + 24 = 37$$
$$(CD)_{22} = (4)(0) + (5)(3) + (6)(1) = 0 + 15 + 6 = 21$$
$$CD = \begin{bmatrix} 16 & 9 \\ 37 & 21 \end{bmatrix}$$
$$C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}_{2 \times 3}, \quad D = \begin{bmatrix} 2 & 0 \\ 1 & 3 \\ 4 & 1 \end{bmatrix}_{3 \times 2}$$
Check: $2 \times 3$ and $3 \times 2$ → result is $2 \times 2$ ✓
$$(CD)_{11} = (1)(2) + (2)(1) + (3)(4) = 2 + 2 + 12 = 16$$
$$(CD)_{12} = (1)(0) + (2)(3) + (3)(1) = 0 + 6 + 3 = 9$$
$$(CD)_{21} = (4)(2) + (5)(1) + (6)(4) = 8 + 5 + 24 = 37$$
$$(CD)_{22} = (4)(0) + (5)(3) + (6)(1) = 0 + 15 + 6 = 21$$
$$CD = \begin{bmatrix} 16 & 9 \\ 37 & 21 \end{bmatrix}$$
Important: Matrix multiplication is NOT commutative!
$AB \neq BA$ (in most cases)
Even when both $AB$ and $BA$ are defined, they usually give different results.
$AB \neq BA$ (in most cases)
Even when both $AB$ and $BA$ are defined, they usually give different results.
Example 3: Order matters
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix}$$
$AB$:
$$AB = \begin{bmatrix} (1)(0)+(2)(2) & (1)(1)+(2)(0) \\ (3)(0)+(4)(2) & (3)(1)+(4)(0) \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 8 & 3 \end{bmatrix}$$
$BA$:
$$BA = \begin{bmatrix} (0)(1)+(1)(3) & (0)(2)+(1)(4) \\ (2)(1)+(0)(3) & (2)(2)+(0)(4) \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ 2 & 4 \end{bmatrix}$$
Notice: $AB \neq BA$
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix}$$
$AB$:
$$AB = \begin{bmatrix} (1)(0)+(2)(2) & (1)(1)+(2)(0) \\ (3)(0)+(4)(2) & (3)(1)+(4)(0) \end{bmatrix} = \begin{bmatrix} 4 & 1 \\ 8 & 3 \end{bmatrix}$$
$BA$:
$$BA = \begin{bmatrix} (0)(1)+(1)(3) & (0)(2)+(1)(4) \\ (2)(1)+(0)(3) & (2)(2)+(0)(4) \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ 2 & 4 \end{bmatrix}$$
Notice: $AB \neq BA$
7. Properties of Matrices
Properties of Addition
Commutative Property of Addition:
$$A + B = B + A$$
Order doesn't matter for addition
Associative Property of Addition:
$$(A + B) + C = A + (B + C)$$
Grouping doesn't matter for addition
Additive Identity:
$$A + O = O + A = A$$
where $O$ is the zero matrix (all elements are 0)
Additive Inverse:
$$A + (-A) = O$$
Adding a matrix to its negative gives zero matrix
$$A + B = B + A$$
Order doesn't matter for addition
Associative Property of Addition:
$$(A + B) + C = A + (B + C)$$
Grouping doesn't matter for addition
Additive Identity:
$$A + O = O + A = A$$
where $O$ is the zero matrix (all elements are 0)
Additive Inverse:
$$A + (-A) = O$$
Adding a matrix to its negative gives zero matrix
Properties of Scalar Multiplication
Distributive Property (scalar over addition):
$$k(A + B) = kA + kB$$
Distributive Property (scalars):
$$(k + l)A = kA + lA$$
Associative Property:
$$k(lA) = (kl)A$$
Identity Property:
$$1 \cdot A = A$$
Multiplying by 1 doesn't change the matrix
$$k(A + B) = kA + kB$$
Distributive Property (scalars):
$$(k + l)A = kA + lA$$
Associative Property:
$$k(lA) = (kl)A$$
Identity Property:
$$1 \cdot A = A$$
Multiplying by 1 doesn't change the matrix
Properties of Matrix Multiplication
NOT Commutative:
$$AB \neq BA$$ (usually)
Order matters!
Associative Property:
$$(AB)C = A(BC)$$
Grouping doesn't matter (when defined)
Distributive Property (left):
$$A(B + C) = AB + AC$$
Distributive Property (right):
$$(A + B)C = AC + BC$$
Identity Property:
$$AI = IA = A$$
where $I$ is the identity matrix
Zero Property:
$$AO = OA = O$$
Multiplying by zero matrix gives zero matrix
$$AB \neq BA$$ (usually)
Order matters!
Associative Property:
$$(AB)C = A(BC)$$
Grouping doesn't matter (when defined)
Distributive Property (left):
$$A(B + C) = AB + AC$$
Distributive Property (right):
$$(A + B)C = AC + BC$$
Identity Property:
$$AI = IA = A$$
where $I$ is the identity matrix
Zero Property:
$$AO = OA = O$$
Multiplying by zero matrix gives zero matrix
Properties Summary
Example: Verify properties
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$
Commutative Addition:
$A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} = B + A$ ✓
Identity Matrix:
$AI = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = A$ ✓
$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$
Commutative Addition:
$A + B = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} = B + A$ ✓
Identity Matrix:
$AI = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = A$ ✓
Matrix Operations Summary
| Operation | Requirements | Formula/Method | Result Dimensions |
|---|---|---|---|
| Addition | Same dimensions | $A + B = [a_{ij} + b_{ij}]$ | Same as originals |
| Subtraction | Same dimensions | $A - B = [a_{ij} - b_{ij}]$ | Same as originals |
| Scalar Multiplication | None | $kA = [k \cdot a_{ij}]$ | Same as original |
| Matrix Multiplication | Cols of A = Rows of B | Row × Column | $m \times p$ (outer dimensions) |
Special Matrices Reference
| Type | Description | Example |
|---|---|---|
| Square Matrix | Rows = Columns | $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ ($2 \times 2$) |
| Row Matrix | 1 row, n columns | $\begin{bmatrix} 5 & 2 & 8 \end{bmatrix}$ ($1 \times 3$) |
| Column Matrix | m rows, 1 column | $\begin{bmatrix} 3 \\ 7 \end{bmatrix}$ ($2 \times 1$) |
| Zero Matrix | All elements = 0 | $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$ |
| Identity Matrix | 1's on diagonal, 0's elsewhere | $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ |
| Diagonal Matrix | Non-zero only on diagonal | $\begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$ |
Properties Quick Reference
| Property | Addition | Scalar Mult. | Matrix Mult. |
|---|---|---|---|
| Commutative | YES: $A+B = B+A$ | N/A | NO: $AB \neq BA$ |
| Associative | YES: $(A+B)+C = A+(B+C)$ | YES: $k(lA) = (kl)A$ | YES: $(AB)C = A(BC)$ |
| Distributive | N/A | YES: $k(A+B) = kA+kB$ | YES: $A(B+C) = AB+AC$ |
| Identity | $A + O = A$ | $1 \cdot A = A$ | $AI = IA = A$ |
Success Tips for Matrices:
✓ Dimensions are always ROWS × COLUMNS (in that order)
✓ For addition/subtraction: dimensions must match exactly
✓ Scalar multiplication: multiply EVERY element
✓ For matrix multiplication: COLUMNS of first = ROWS of second
✓ Matrix multiplication: Row × Column (not element-by-element)
✓ Order matters for multiplication: $AB \neq BA$
✓ Element notation: $a_{ij}$ means row $i$, column $j$
✓ Identity matrix has 1's on diagonal, 0's elsewhere
✓ Do scalar multiplication BEFORE addition/subtraction
✓ Practice checking dimensions before performing operations!
✓ Dimensions are always ROWS × COLUMNS (in that order)
✓ For addition/subtraction: dimensions must match exactly
✓ Scalar multiplication: multiply EVERY element
✓ For matrix multiplication: COLUMNS of first = ROWS of second
✓ Matrix multiplication: Row × Column (not element-by-element)
✓ Order matters for multiplication: $AB \neq BA$
✓ Element notation: $a_{ij}$ means row $i$, column $j$
✓ Identity matrix has 1's on diagonal, 0's elsewhere
✓ Do scalar multiplication BEFORE addition/subtraction
✓ Practice checking dimensions before performing operations!