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Lesson 1: Matrix Operations

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Learning Objectives

Understand matrix notation, dimensions, and terminology
Identify matrix entries using row and column notation
Determine when two matrices are equal
Add and subtract matrices with compatible dimensions
Multiply matrices by scalars
Multiply matrices using the row-by-column method
Recognize when matrix operations are defined
Apply properties of matrix operations
Identify and work with special matrices (zero and identity matrices)
Apply matrices to real-world problems in economics, transformations, and networks

Introduction: What is a Matrix?

Matrix: A rectangular array of numbers arranged in rows and columns, enclosed in brackets.

Matrices (plural of matrix) are powerful tools in mathematics used to organize data, solve systems of equations, perform transformations, and model real-world situations. They appear in computer graphics, economics, engineering, statistics, and many other fields.

Example 1: Matrix Notation

Here are examples of matrices with different sizes:

2 × 3 matrix (2 rows, 3 columns):

1	2	3
4	5	6

3 × 2 matrix (3 rows, 2 columns):

7	8
9	10
11	12

1 × 4 matrix (row matrix):

-3

3 × 1 matrix (column matrix):

-2

Matrix Dimensions: A matrix with m rows and n columns is called an m × n matrix (read "m by n"). The dimensions are always written as rows × columns.

Entry notation: The entry in row i and column j is denoted a_ij. For example, a₂₃ is the entry in row 2, column 3.

Example 2: Identifying Matrix Entries

Consider the matrix A =

5	-2	7
1	0	-3
4	6	2

Solution:

This is a 3 × 3 matrix. Let's identify some entries:

a₁₁ = 5 (row 1, column 1)
a₁₂ = -2 (row 1, column 2)
a₂₃ = -3 (row 2, column 3)
a₃₂ = 6 (row 3, column 2)
a₃₃ = 2 (row 3, column 3)

Example 3: Determining Matrix Dimensions

State the dimensions of each matrix:

(a) B =

1	0
0	1

Answer: 2 × 2 (square matrix)

(b) C =

-1

Answer: 1 × 4 (row matrix)

2	5	-1
0	3	7
1	-2	4
6	8	-3

Answer: 4 × 3

Section 1: Matrix Equality

Equal Matrices: Two matrices are equal if and only if:

They have the same dimensions, AND
All corresponding entries are equal

Example 4: Determining Matrix Equality

Are these matrices equal?

A =

2	5
-1	3

and B =

2	5
-1	3

Solution:

Both are 2 × 2 matrices and all corresponding entries are equal:

a₁₁ = b₁₁ = 2
a₁₂ = b₁₂ = 5
a₂₁ = b₂₁ = -1
a₂₂ = b₂₂ = 3

Answer: Yes, A = B

Example 5: Finding Unknown Entries

Find x and y if

2x	5
3	y + 1

8	5
3	7

Solution:

For the matrices to be equal, corresponding entries must be equal:

From entry (1,1): 2x = 8, so x = 4
From entry (2,2): y + 1 = 7, so y = 6

Answer: x = 4, y = 6

Example 6: Different Dimensions

Can these matrices be equal?

A =

and B =

Solution:

A is 1 × 3 and B is 3 × 1. Since they have different dimensions, they cannot be equal, even though they contain the same numbers.

Answer: No, they are not equal.

Section 2: Matrix Addition and Subtraction

Matrix Addition: If A and B are matrices of the same dimensions, then A + B is the matrix obtained by adding corresponding entries.

If A = [a_ij] and B = [b_ij], then (A + B)_ij = a_ij + b_ij

Important

Matrix addition is only defined for matrices with the SAME dimensions. You cannot add a 2 × 3 matrix to a 3 × 2 matrix.

Example 7: Adding Matrices

Find A + B where A =

2	5
-1	3

and B =

4	-2
0	7

Solution:

Both matrices are 2 × 2, so addition is defined. Add corresponding entries:

A + B =

2+4	5+(-2)
-1+0	3+7

6	3
-1	10

Answer: A + B =

6	3
-1	10

Matrix Subtraction: If A and B are matrices of the same dimensions, then A - B is the matrix obtained by subtracting corresponding entries.

If A = [a_ij] and B = [b_ij], then (A - B)_ij = a_ij - b_ij

Example 8: Subtracting Matrices

Find C - D where C =

5	-3	2
1	0	-4

and D =

2	1	-1
3	-2	5

Solution:

Both are 2 × 3 matrices, so subtraction is defined:

C - D =

5-2	-3-1	2-(-1)
1-3	0-(-2)	-4-5

3	-4	3
-2	2	-9

Example 9: Incompatible Dimensions

Can we add these matrices?

E =

1	2
3	4

and F =

5	6	7
8	9	10

Solution:

E is 2 × 2 and F is 2 × 3. They have different dimensions.

Answer: No, E + F is undefined because the matrices have different dimensions.

Example 10: Combined Operations

If A =

1	2
3	4

, B =

5	-1
2	0

, and C =

-2	3
1	-5

, find A + B - C.

Solution:

Work from left to right:

A + B =

6	1
5	4

(A + B) - C =

6	1
5	4

-2	3
1	-5

8	-2
4	9

Section 3: Scalar Multiplication

Scalar Multiplication: If A is a matrix and k is a real number (called a scalar), then kA is the matrix obtained by multiplying every entry of A by k.

If A = [a_ij], then (kA)_ij = k · a_ij

Example 11: Multiplying by a Scalar

Find 3A where A =

2	-1
4	5

Solution:

Multiply each entry by 3:

3A =

3(2)	3(-1)
3(4)	3(5)

6	-3
12	15

Example 12: Negative Scalar

Find -2B where B =

3	0	-4
1	-2	5

Solution:

-2B =

-6	0	8
-2	4	-10

Example 13: Fractional Scalar

Find (1/2)C where C =

8	-6
4	10
-2	0

Solution:

(1/2)C =

4	-3
2	5
-1	0

Example 14: Combining Scalar Multiplication and Addition

If A =

1	2
3	4

and B =

5	6
7	8

, find 2A + 3B.

Solution:

Step 1: Find 2A:

2A =

2	4
6	8

Step 2: Find 3B:

3B =

15	18
21	24

Step 3: Add the results:

2A + 3B =

17	22
27	32

Section 4: Matrix Multiplication

Matrix multiplication is more complex than addition or scalar multiplication. It is NOT done by simply multiplying corresponding entries.

Matrix Multiplication: If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix.

Key requirement: The number of columns in A must equal the number of rows in B.

The entry in row i, column j of AB is found by multiplying the entries in row i of A by the corresponding entries in column j of B, then adding the products.

Row × Column Method:

To find entry (i,j) in the product AB:

Take row i from matrix A
Take column j from matrix B
Multiply corresponding entries and add the results

Dimension Compatibility

For AB to be defined, the dimensions must satisfy:

A is (m × n) and B is (n × p) → AB is (m × p)

The inner dimensions (both n) must match. The outer dimensions (m and p) give the size of the product.

Example 15: Basic Matrix Multiplication (2 × 2)

Find AB where A =

2	3
1	4

and B =

5	6
7	8

Solution:

A is 2 × 2 and B is 2 × 2, so AB is defined and will be 2 × 2.

Entry (1,1): Row 1 of A × Column 1 of B

[2, 3] · [5, 7]^T = 2(5) + 3(7) = 10 + 21 = 31

Entry (1,2): Row 1 of A × Column 2 of B

[2, 3] · [6, 8]^T = 2(6) + 3(8) = 12 + 24 = 36

Entry (2,1): Row 2 of A × Column 1 of B

[1, 4] · [5, 7]^T = 1(5) + 4(7) = 5 + 28 = 33

Entry (2,2): Row 2 of A × Column 2 of B

[1, 4] · [6, 8]^T = 1(6) + 4(8) = 6 + 32 = 38

Answer: AB =

31	36
33	38

Example 16: Different Dimensions

Find AB where A =

1	2	3
4	5	6

and B =

7	8
9	10
11	12

Solution:

A is 2 × 3 and B is 3 × 2. The product AB will be 2 × 2.

Entry (1,1):

[1, 2, 3] · [7, 9, 11]^T = 1(7) + 2(9) + 3(11) = 7 + 18 + 33 = 58

Entry (1,2):

[1, 2, 3] · [8, 10, 12]^T = 1(8) + 2(10) + 3(12) = 8 + 20 + 36 = 64

Entry (2,1):

[4, 5, 6] · [7, 9, 11]^T = 4(7) + 5(9) + 6(11) = 28 + 45 + 66 = 139

Entry (2,2):

[4, 5, 6] · [8, 10, 12]^T = 4(8) + 5(10) + 6(12) = 32 + 50 + 72 = 154

Answer: AB =

58	64
139	154

Example 17: Matrix Multiplication is Not Commutative

Given A =

1	2
3	4

and B =

0	5
6	7

, show that AB ≠ BA.

Solution:

Calculate AB:

AB =

1(0)+2(6)	1(5)+2(7)
3(0)+4(6)	3(5)+4(7)

12	19
24	43

Calculate BA:

BA =

0(1)+5(3)	0(2)+5(4)
6(1)+7(3)	6(2)+7(4)

15	20
27	40

Answer: AB ≠ BA. Matrix multiplication is NOT commutative.

Example 18: When Multiplication is Undefined

Can we multiply these matrices?

C =

1	2
3	4

and D =

5	6	7
8	9	10
11	12	13

Solution:

C is 2 × 2 and D is 3 × 3.

For CD: C has 2 columns but D has 3 rows. The inner dimensions don't match.

Answer: CD is undefined. (Note: DC would be defined and would result in a 3 × 2 matrix.)

Example 19: Row Matrix Times Column Matrix

Find AB where A =

-3

and B =

-2

Solution:

A is 1 × 3 and B is 3 × 1, so AB is 1 × 1 (a single number).

AB = [2(-1) + (-3)(5) + 4(-2)] = [2 - 15 - 8] = [-21]

Or simply: AB = -21

Example 20: Larger Matrix Multiplication

Find AB where A =

1	0	2
-1	3	1

and B =

3	1
2	1
1	0

Solution:

A is 2 × 3, B is 3 × 2, so AB is 2 × 2.

AB =

1(3)+0(2)+2(1)	1(1)+0(1)+2(0)
-1(3)+3(2)+1(1)	-1(1)+3(1)+1(0)

5	1
4	2

Section 5: Properties of Matrix Operations

Properties of Matrix Addition and Scalar Multiplication:

For matrices A, B, C (of the same dimensions) and scalars k, m:

Commutative: A + B = B + A
Associative: (A + B) + C = A + (B + C)
Zero Matrix: A + O = A (where O is the zero matrix)
Additive Inverse: A + (-A) = O
Scalar Distribution: k(A + B) = kA + kB
Scalar Distribution: (k + m)A = kA + mA
Scalar Association: k(mA) = (km)A
Identity: 1 · A = A

Properties of Matrix Multiplication:

For matrices A, B, C (where products are defined) and scalar k:

Associative: (AB)C = A(BC)
Left Distributive: A(B + C) = AB + AC
Right Distributive: (A + B)C = AC + BC
Scalar Association: k(AB) = (kA)B = A(kB)
NOT Commutative: In general, AB ≠ BA

Example 21: Verifying Distributive Property

Verify that A(B + C) = AB + AC for:

A =

2	1
3	4

, B =

1	0
2	3

, C =

4	5
6	7

Solution:

Left side: A(B + C)

First: B + C =

5	5
8	10

Then: A(B + C) =

18	20
47	55

Right side: AB + AC

AB =

4	3
11	12

AC =

14	17
36	43

AB + AC =

18	20
47	55

Answer: Both sides equal

18	20
47	55

. Property verified!

Section 6: Special Matrices

Zero Matrix (O): A matrix where all entries are zero.

Example: The 2 × 3 zero matrix is

0	0	0
0	0	0

Identity Matrix (I): A square matrix with 1's on the main diagonal and 0's elsewhere.

The identity matrix acts like the number 1 in matrix multiplication: AI = IA = A

Example 22: Identity Matrices

Write the 2 × 2 and 3 × 3 identity matrices.

Solution:

2 × 2 identity: I₂ =

1	0
0	1

3 × 3 identity: I₃ =

1	0	0
0	1	0
0	0	1

Example 23: Identity Property

Verify that AI = A for A =

2	5
3	7

Solution:

AI =

2	5
3	7

1	0
0	1

2(1)+5(0)	2(0)+5(1)
3(1)+7(0)	3(0)+7(1)

2	5
3	7

= A

Section 7: Applications of Matrices

Example 24: Inventory Application

A company has two stores. Store 1 has 30 laptops, 50 tablets, and 20 smartphones. Store 2 has 25 laptops, 40 tablets, and 35 smartphones. Write this information as a matrix. If each laptop costs $800, each tablet costs $400, and each smartphone costs $600, use matrix multiplication to find the total inventory value at each store.

Solution:

Inventory matrix:

A =

30	50	20
25	40	35

(rows: stores; columns: product types)

Price column matrix:

P =

800

400

600

Calculate AP:

AP =

30(800)+50(400)+20(600)

25(800)+40(400)+35(600)

56,000

57,000

Answer: Store 1 has $56,000 in inventory; Store 2 has $57,000 in inventory.

Example 25: Transformation Application

A transformation matrix T =

2	0
0	2

scales points by a factor of 2. Apply this transformation to the point (3, 4).

Solution:

Represent the point as a column matrix:

P =

Multiply by the transformation matrix:

TP =

2	0
0	2

Answer: The transformed point is (6, 8), which is (3, 4) scaled by a factor of 2.

Check Your Understanding

1. What are the dimensions of the matrix

1	2	3	4
5	6	7	8

Answer: 2 × 4 (2 rows, 4 columns)

2. Find A + B where A =

3	-1
2	5

and B =

4	2
-3	1

Answer:

7	1
-1	6

3. Find 4C where C =

2	-3
1	0

Answer:

8	-12
4	0

4. If A is 3 × 4 and B is 4 × 2, what are the dimensions of AB?

Answer: 3 × 2 (outer dimensions)

5. Can you multiply a 2 × 3 matrix by a 2 × 3 matrix?

Answer: No. The first matrix has 3 columns but the second has 2 rows. The inner dimensions don't match.

6. Find AB where A =

1	2
3	4

and B =

2	0
1	3

Answer:

4	6
10	12

Entry (1,1): 1(2) + 2(1) = 4; Entry (1,2): 1(0) + 2(3) = 6; Entry (2,1): 3(2) + 4(1) = 10; Entry (2,2): 3(0) + 4(3) = 12

7. Is matrix multiplication commutative? That is, does AB always equal BA?

Answer: No, matrix multiplication is NOT commutative. In general, AB ≠ BA.

8. What is the 2 × 2 identity matrix?

Answer:

1	0
0	1

9. Find 2A - 3B where A =

5	1
2	-3

and B =

2	4
1	-1

Answer:

4	-10
1	-3

2A =

10	2
4	-6

; 3B =

6	12
3	-3

; 2A - 3B =

4	-10
1	-3

10. Find [2, -1, 3] ·

Answer: 21

2(4) + (-1)(5) + 3(6) = 8 - 5 + 18 = 21

Key Takeaways

A matrix is a rectangular array of numbers with dimensions m × n (rows × columns)
Entry a_ij is in row i, column j
Two matrices are equal if they have the same dimensions and all corresponding entries are equal
Matrix addition and subtraction: Add or subtract corresponding entries (only defined for same dimensions)
Scalar multiplication: Multiply every entry by the scalar
Matrix multiplication: Use row × column method; (AB)_ij = row i of A × column j of B
For AB to be defined: number of columns in A must equal number of rows in B
If A is m × n and B is n × p, then AB is m × p
Matrix multiplication is NOT commutative: AB ≠ BA in general
Matrix multiplication IS associative: (AB)C = A(BC)
Distributive property: A(B + C) = AB + AC
Zero matrix (O): All entries are 0; A + O = A
Identity matrix (I): 1's on diagonal, 0's elsewhere; AI = IA = A
Matrices are used for data organization, transformations, economics, and solving systems

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