How to quickly determine if a matrix is invertible?

Matrix A is invertible when det(A)≠0.

Generally no, matrix multiplication is not commutative.

How to solve Ax=b using matrix inverse?

If A is invertible, then x=A^{-1}b.

Lesson 2-1: Matrices & Inverses

Matrices & Inverses

Master fundamental matrix operations including addition, multiplication, determinants, and inverses. Learn to solve linear systems using matrix methods and apply these concepts to real-world problems in computer graphics, economics, and engineering.

Learning Objectives

Matrix Operations

Master addition, scalar multiplication, and matrix multiplication

Determinants

Calculate determinants for 2×2 and 3×3 matrices

Matrix Inverses

Find inverses of 2×2 matrices and solve linear systems

Real-World Applications

Apply matrices to computer graphics, economics, and engineering

Core Knowledge Points

Matrix Operations

Matrix Addition

$A + B = [a_{ij} + b_{ij}]$ (same dimensions required)

Add corresponding elements

Scalar Multiplication

$kA = [ka_{ij}]$

Multiply each element by the scalar

Matrix Multiplication

$(AB)_{ij} = \sum_{k} a_{ik} b_{kj}$

Number of columns in A = number of rows in B

Determinants

2×2 Determinant

$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$

3×3 Determinant (Sarrus Rule)

$\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh$

Matrix Inverses

2×2 Inverse Formula

$A^{-1} = \frac{1}{\det A} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Only exists when $\det A \neq 0$

Solving Linear Systems

$Ax = b \Rightarrow x = A^{-1}b$

When A is invertible

Real-World Applications

Computer Graphics & Engineering

2D Transformations

Rotation matrix: $R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

Used in computer graphics for rotating objects

Economic Models

Input-output models: $x = (I - A)^{-1}d$

Where A is the technology matrix, d is demand vector

Comprehensive Example Analysis

Matrix Operations Example

Given: $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}$

1. Matrix Addition

$A + B = \begin{pmatrix} 2+1 & 3+2 \\ 1+3 & 4+1 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 4 & 5 \end{pmatrix}$

2. Matrix Multiplication

$AB = \begin{pmatrix} 2(1)+3(3) & 2(2)+3(1) \\ 1(1)+4(3) & 1(2)+4(1) \end{pmatrix} = \begin{pmatrix} 11 & 7 \\ 13 & 6 \end{pmatrix}$

3. Determinant of A

$\det A = 2(4) - 3(1) = 8 - 3 = 5$

4. Inverse of A

$A^{-1} = \frac{1}{5}\begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix} = \begin{pmatrix} \frac{4}{5} & -\frac{3}{5} \\ -\frac{1}{5} & \frac{2}{5} \end{pmatrix}$

Practice Problems

Problem 1: Matrix Operations

Given $A = \begin{pmatrix} 3 & -1 \\ 2 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ -1 & 3 \end{pmatrix}$ :

a) Calculate A + B

b) Calculate AB

c) Calculate BA

d) Find det(A) and det(B)

Problem 2: Matrix Inverses

For the matrix $C = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}$ :

a) Find C⁻¹

b) Verify that CC⁻¹ = I

c) Solve the system Cx = b where b = (5, 7)ᵀ

Problem 3: Real-World Application

A computer graphics system needs to rotate a point (3, 4) by 45° counterclockwise:

a) Write the rotation matrix for 45°

b) Calculate the new coordinates

c) Verify the result using trigonometry

Key Takeaways

Matrix Operations

Follow specific rules for addition, multiplication, and scalar operations

Determinants

Calculate using formulas and understand their geometric meaning

Matrix Inverses

Use to solve linear systems and understand when they exist

Applications

Essential for computer graphics, economics, and engineering

Advanced Insights

Matrix Multiplication Properties: Matrix multiplication is associative but not commutative: (AB)C = A(BC) but AB ≠ BA in general.

Determinant Properties: det(AB) = det(A)det(B) and det(A⁻¹) = 1/det(A) when A is invertible.

Linear Independence: A matrix is invertible if and only if its columns are linearly independent.

Common Pitfalls

• Confusing matrix multiplication with element-wise multiplication
• Forgetting that matrix multiplication is not commutative
• Not checking if a matrix is invertible before trying to find its inverse
• Making arithmetic errors when calculating determinants

Step-by-Step Guides

Find 2×2 Inverse

Compute $d=ad-bc$ for $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ ;
If $d\ne0$ , write $A^{-1}=\tfrac{1}{d}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$ ;
Verify by checking $AA^{-1}=I$ and $A^{-1}A=I$ .

Determinant by Row Operations (3×3)

Swap rows flips sign of determinant
Scale a row scales determinant by same scalar
Add multiple of a row to another leaves determinant unchanged

Solving Ax=b via Inverse

Check invertibility: $\det(A) \ne 0$ ;
Compute $A^{-1}$ or use a factorization method;
Compute $x=A^{-1}b$ and interpret the solution.

Glossary

Identity Matrix: Square matrix with ones on diagonal and zeros elsewhere; neutral element for multiplication.
Singular: Non-invertible matrix with determinant 0; maps space to lower dimension.
Adjugate: Transpose of cofactor matrix; used in formula for inverse.
Row-Echelon Form: Upper triangular-like form obtained by elementary row operations.

Additional Practice

Set A: Determinants

A1. Compute det of random 3×3 with two identical rows (expect 0)
A2. Show det(AB)=det(A)det(B) for 2×2 by direct computation
A3. Use row ops to compute det of sparse 4×4

Set B: Inverses

B1. Decide invertibility by determinant value
B2. Find inverse via adjugate method for 3×3 with zeros
B3. Verify A^-1 A = I numerically

Insights & Tips

• When columns are independent, transformations preserve dimension and are invertible.
• Determinant magnitude measures area/volume scaling; sign encodes orientation.
• Use LU decomposition to speed repeated solves with same A and varying b.

Quick Checks

Is A invertible if det(A)=0?

No. det(A)=0 → A not invertible.

Does AB=BA hold?

Generally no. Only for special commuting pairs.

When is A^-1b preferred?

Rarely in computation; use factorization like LU.

Mini Projects

Image Transformations: Apply rotation and scaling matrices to pixel coordinates (theoretical pipeline).
Economics Input-Output: Analyze small 2-sector model with $x=(I-A)^{-1}d$ and interpret sensitivity.
Robotics Kinematics: Compose planar transforms and discuss invertibility of poses.

Summary Checklist

[ ] I can compute 2×2, 3×3 determinants
[ ] I can decide invertibility and find A^-1 for 2×2
[ ] I can solve Ax=b via inverse conceptually and via elimination practically
[ ] I can explain geometric meaning of determinant

← Back to Unit 2 Next: Systems & Linear Combinations →

Matrices & Inverses

Learning Objectives

Matrix Operations

Determinants

Matrix Inverses

Real-World Applications

Core Knowledge Points

Matrix Operations

Matrix Addition

Scalar Multiplication

Matrix Multiplication

Determinants

2×2 Determinant

3×3 Determinant (Sarrus Rule)

Matrix Inverses

2×2 Inverse Formula

Solving Linear Systems

Real-World Applications

Computer Graphics & Engineering

2D Transformations

Economic Models

Comprehensive Example Analysis

Matrix Operations Example

Given: A=(2314)A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}A=(21​34​), B=(1231)B = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}B=(13​21​)

1. Matrix Addition

2. Matrix Multiplication

3. Determinant of A

4. Inverse of A

Practice Problems

Problem 1: Matrix Operations

Problem 2: Matrix Inverses

Problem 3: Real-World Application

Key Takeaways

Matrix Operations

Determinants

Matrix Inverses

Applications

Advanced Insights

Common Pitfalls

Step-by-Step Guides

Find 2×2 Inverse

Determinant by Row Operations (3×3)

Solving Ax=b via Inverse

Glossary

Additional Practice

Set A: Determinants

Set B: Inverses

Insights & Tips

Quick Checks

Mini Projects

Summary Checklist

Given: $A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 2 \\ 3 & 1 \end{pmatrix}$