Determinants and Matrix Inverses

What is a Determinant?

Determinant: Single number associated with square matrix

Notation: det(A) or |A|

Properties:

• Only defined for square matrices
• Indicates if matrix is invertible
• Related to area/volume transformations

Key fact: det(A) ≠ 0 means A is invertible

Determinant of 2×2 Matrix

Formula:

|a  b|
|c  d| = ad - bc

Remember: Diagonal products (main diagonal minus other diagonal)

Example 1: Calculate 2×2 Determinant

Matrix:

|3  2|
|1  4|

Calculate:

det = 3(4) - 2(1)
    = 12 - 2
    = 10

Example 2: Zero Determinant

Matrix:

|2  4|
|3  6|

Calculate:

det = 2(6) - 4(3)
    = 12 - 12
    = 0

det = 0 means matrix is NOT invertible (singular)

Determinant of 3×3 Matrix

Method 1: Cofactor expansion

Formula (expanding along first row):

|a  b  c|
|d  e  f| = a|e f| - b|d f| + c|d e|
|g  h  i|     |h i|     |g i|     |g h|

Pattern: Alternate signs (+, -, +)

Example: 3×3 Determinant

Matrix:

|2  1  3|
|0  4  1|
|5  2  6|

Expand along first row:

det = 2|4 1| - 1|0 1| + 3|0 4|
       |2 6|     |5 6|     |5 2|

    = 2(4·6 - 1·2) - 1(0·6 - 1·5) + 3(0·2 - 4·5)
    = 2(24 - 2) - 1(0 - 5) + 3(0 - 20)
    = 2(22) - 1(-5) + 3(-20)
    = 44 + 5 - 60
    = -11

Properties of Determinants

1. Identity matrix: det(I) = 1

2. Transpose: det(Aᵀ) = det(A)

3. Product: det(AB) = det(A) · det(B)

4. Scalar multiple: det(kA) = kⁿ · det(A) for n×n matrix

5. Row swap: Changes sign of determinant

6. Row of zeros: det = 0

7. Proportional rows: det = 0

Example: Row Operations

Original:

|2  1|
|4  3| → det = 6 - 4 = 2

Swap rows:

|4  3|
|2  1| → det = 4 - 6 = -2

Sign changed!

What is an Inverse Matrix?

Inverse matrix A⁻¹: Matrix where AA⁻¹ = A⁻¹A = I

Like reciprocal for numbers: 5 × (1/5) = 1

Requirements:

• Matrix must be square
• det(A) ≠ 0

If det(A) = 0: Matrix has no inverse (singular)

Finding 2×2 Inverse

Formula:

If A = [a  b]
       [c  d]

Then A⁻¹ = (1/det(A)) [d  -b]
                       [-c  a]

Steps:

1. Calculate det(A) = ad - bc
2. Swap main diagonal elements (a ↔ d)
3. Negate other diagonal (b and c)
4. Multiply by 1/det(A)

Example 1: Find 2×2 Inverse

Matrix:

A = [3  2]
    [1  4]

Step 1: Determinant

det = 3(4) - 2(1) = 10

Step 2: Apply formula

A⁻¹ = (1/10) [4  -2]
             [-1  3]

    = [0.4  -0.2]
      [-0.1  0.3]

Verify: AA⁻¹ = I

[3  2] [0.4  -0.2]   [1  0]
[1  4] [-0.1  0.3] = [0  1] ✓

Example 2: No Inverse

Matrix:

A = [2  4]
    [1  2]

Determinant:

det = 2(2) - 4(1) = 0

No inverse exists! (Rows are proportional)

Finding 3×3 Inverse

Methods:

1. Adjugate matrix method
2. Gaussian elimination (augment with I)
3. Calculator/software

Formula: A⁻¹ = (1/det(A)) · adj(A)

For this level, use technology or provided formula

Example: 3×3 Inverse (Given)

Matrix:

A = [1  2  0]
    [0  1  3]
    [2  0  1]

det(A) = -13 (calculated)

Inverse (using formula):

A⁻¹ = [1/13   -2/13   6/13]
      [6/13   1/13   -3/13]
      [-2/13  4/13   1/13]

Solving Systems with Inverse Matrices

Matrix equation: AX = B

Solution: X = A⁻¹B

Advantage: Efficient for multiple systems with same coefficients

Example: Solve System

System:

3x + 2y = 7
x + 4y = 10

Matrix form:

[3  2] [x]   [7]
[1  4] [y] = [10]

From previous example:

A⁻¹ = [0.4  -0.2]
      [-0.1  0.3]

Solve:

[x]   [0.4  -0.2] [7]    [0.4(7) - 0.2(10)]   [0.8]
[y] = [-0.1  0.3] [10] = [-0.1(7) + 0.3(10)] = [2.3]

Solution: x = 0.8, y = 2.3

Check:

3(0.8) + 2(2.3) = 2.4 + 4.6 = 7 ✓
0.8 + 4(2.3) = 0.8 + 9.2 = 10 ✓

Applications: Cryptography

Hill cipher: Uses matrix multiplication to encode messages

Encoding: C = MK (mod 26)

• M = message matrix
• K = key matrix
• C = coded matrix

Decoding: M = CK⁻¹ (mod 26)

Requires: det(K) ≠ 0 and relatively prime to 26

Example: Simple Encoding

Message: "HI" → [7, 8] (H=7, I=8 in 0-25 system)

Key matrix:

K = [3  2]
    [1  4]

Encode:

C = [7  8] [3  2] = [21 + 8   14 + 32]
           [1  4]   = [29      46]

Reduce mod 26:

[3  20] → "DU"

To decode, recipient uses K⁻¹

Determinant and Area/Volume

2D: |det(A)| = area scaling factor

3D: |det(A)| = volume scaling factor

Negative determinant: Orientation reverses

Example: Area Transformation

Transformation matrix:

A = [2  0]
    [0  3]

det(A) = 6

Unit square (area 1) transforms to rectangle with area 6

Stretches by 2 in x-direction, 3 in y-direction

Cramer's Rule

Method to solve systems using determinants

For AX = B:

x₁ = det(A₁)/det(A)
x₂ = det(A₂)/det(A)
...

Where Aᵢ = A with column i replaced by B

Example: Cramer's Rule

System:

2x + y = 5
x + 3y = 8

Coefficient matrix:

A = [2  1]
    [1  3]

det(A) = 6 - 1 = 5

For x, replace first column with constants:

A₁ = [5  1]
     [8  3]

det(A₁) = 15 - 8 = 7
x = 7/5 = 1.4

For y, replace second column:

A₂ = [2  5]
     [1  8]

det(A₂) = 16 - 5 = 11
y = 11/5 = 2.2

Solution: (1.4, 2.2)

Applications: Computer Graphics

Transformations represented by matrices:

• Rotation
• Scaling
• Reflection
• Shear

Inverse matrices: Undo transformations

det = 0: Transformation collapses to lower dimension

Example: Rotation Matrix

Rotate 90° counterclockwise:

R = [0  -1]
    [1   0]

det(R) = 0 - (-1) = 1

Preserves area (det = ±1)

Inverse rotates back:

R⁻¹ = [0   1]  (90° clockwise)
      [-1  0]

Applications: Economics (Leontief Model)

Input-output model: Industries depend on each other

Equation: X = AX + D

• X = production levels
• A = input-output matrix
• D = final demand

Solution: X = (I - A)⁻¹D

Requires: det(I - A) ≠ 0

Identity and Inverse Properties

AA⁻¹ = A⁻¹A = I

(AB)⁻¹ = B⁻¹A⁻¹ (reverse order!)

(A⁻¹)⁻¹ = A

(Aᵀ)⁻¹ = (A⁻¹)ᵀ

det(A⁻¹) = 1/det(A)

When Inverse Doesn't Exist

Matrix is singular (non-invertible) when:

• det(A) = 0
• Rows/columns are linearly dependent
• Transformation collapses dimension

Consequences:

• System may have no solution or infinite solutions
• Cannot use inverse method to solve

Technology Tools

Calculators: TI-84 has matrix operations

Computer software:

• MATLAB, Octave
• Python (NumPy)
• Mathematica, Maple
• R

Spreadsheets: Excel has matrix functions (MINVERSE, MMULT)

Practice