Matrices

📐 Introduction to Matrices

Part 1 of 7

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

$A = \begin{bmatrix} 2 & 3 \\ 1 & -4 \end{bmatrix}$

This is a 2×2 matrix (2 rows, 2 columns).

Notation

• $a_{ij}$ = element in row $i$, column $j$
• $A_{2×3}$ = matrix with 2 rows, 3 columns
• Two matrices are equal if same dimensions AND all corresponding entries match

Special Matrices

➕ Matrix Addition & Scalar Multiplication

Addition (same dimensions required!)

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$

Add corresponding entries.

Scalar Multiplication

$3 \begin{bmatrix} 2 & -1 \\ 0 & 4 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ 0 & 12 \end{bmatrix}$

Multiply every entry by the scalar.

Properties

• $A + B = B + A$ (commutative)
• $(A + B) + C = A + (B + C)$ (associative)
• $k(A + B) = kA + kB$ (distributive)
• $A + O = A$ (additive identity)

💡 You CANNOT add matrices of different dimensions.

📊 Matrices & Systems

Augmented Matrix

Write a system as a matrix:

$\begin{cases} 2x + 3y = 7 \\ x - y = 1 \end{cases} \implies \left[\begin{array}{cc|c} 2 & 3 & 7 \\ 1 & -1 & 1 \end{array}\right]$

The vertical line separates coefficients from constants.

Coefficient Matrix

$A = \begin{bmatrix} 2 & 3 \\ 1 & -1 \end{bmatrix}, \quad \vec{x} = \begin{bmatrix} x \\ y \end{bmatrix}, \quad \vec{b} = \begin{bmatrix} 7 \\ 1 \end{bmatrix}$

$A\vec{x} = \vec{b}$

This compact notation represents the entire system!

Matrix Basics Quiz 🎯

Matrix Arithmetic 🧮

$A = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 5 \\ -3 & 2 \end{bmatrix}$

$A + B = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}$

1) Top-left entry = ?

2) Top-right entry = ?

3) $3A$: top-left entry = ?

Matrix Concepts 🔽

Exit Quiz ✅

✖️ Matrix Multiplication

Part 2 of 7

The Rule: Row × Column

To multiply $A \cdot B$:

• $A$ must have same number of columns as $B$ has rows
• Result dimensions: (rows of $A$) × (columns of $B$)

$A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$

How to Compute Entry $c_{ij}$

Dot product of row $i$ of $A$ with column $j$ of $B$:

$c_{ij} = \sum_{k=1}^n a_{ik} \cdot b_{kj}$

Example

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 1(5)+2(7) & 1(6)+2(8) \\ 3(5)+4(7) & 3(6)+4(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

⚠️ Important Properties

NOT Commutative!

$AB \neq BA \text{ (in general)}$

Example: $AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$ but $BA = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$

Other Properties

• $(AB)C = A(BC)$ — associative ✓
• $A(B+C) = AB+AC$ — distributive ✓
• $AI = IA = A$ — identity ✓
• $kA \cdot B = k(AB) = A \cdot kB$ — scalar ✓

Dimension Check

$A$ is $2 \times 3$, $B$ is $3 \times 4$:

• $AB$: ✅ ($2 \times 4$ result)
• $BA$: ❌ ($4 \neq 2$, can't multiply)

💡 Memory trick: inner dimensions must match, outer dimensions give result size.

🎯 Special Multiplications

Matrix × Column Vector

$\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 2x+y \\ 3x-y \end{bmatrix}$

This is how $A\vec{x} = \vec{b}$ works!

Powers of Matrices

$A^2 = A \cdot A, \quad A^3 = A \cdot A \cdot A$

Only defined for square matrices.

The Identity Matrix

$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

$AI = IA = A$ for any compatible matrix $A$.

Like multiplying by 1 in regular arithmetic!

Multiplication Quiz 🎯

Compute 🧮

$\begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 4 \\ 2 & -1 \end{bmatrix}$

1) Entry $(1,1)$ = ?

2) Entry $(1,2)$ = ?

3) Entry $(2,1)$ = ?

Multiplication Rules 🔽

Exit Quiz ✅

🔢 Determinants

Part 3 of 7

2×2 Determinant

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

Example

$\det \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} = 3(4) - 2(1) = 10$

What Does It Mean?

• $\det \neq 0$: matrix is invertible, system has unique solution
• $\det = 0$: matrix is singular, system is dependent or inconsistent
• $|\det|$ = area of parallelogram formed by row/column vectors

Geometric Interpretation

Rows $\begin{bmatrix} 3 & 2 \end{bmatrix}$ and $\begin{bmatrix} 1 & 4 \end{bmatrix}$ form a parallelogram with area $= |10| = 10$.

If det = 0, the vectors are parallel (linearly dependent).

📊 3×3 Determinant

Expansion Along Row 1 (Cofactor Expansion)

$\det \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)$

Example

$\det \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{bmatrix}$

$= 1(5 \cdot 0 - 6 \cdot 8) - 2(4 \cdot 0 - 6 \cdot 7) + 3(4 \cdot 8 - 5 \cdot 7)$

$= 1(-48) - 2(-42) + 3(-3)$

$= -48 + 84 - 9 = 27$

Sign Pattern for Cofactors

$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$

Checkerboard pattern starting with $+$ at $(1,1)$.

📐 Cramer's Rule

For 2×2 Systems

$\begin{cases} ax+by=e \\ cx+dy=f \end{cases}$

$x = \frac{\det \begin{bmatrix} e & b \\ f & d \end{bmatrix}}{\det \begin{bmatrix} a & b \\ c & d \end{bmatrix}}, \quad y = \frac{\det \begin{bmatrix} a & e \\ c & f \end{bmatrix}}{\det \begin{bmatrix} a & b \\ c & d \end{bmatrix}}$

Example: $\begin{cases} 3x+2y=8 \\ x-y=1 \end{cases}$

$D = 3(-1)-2(1) = -5$

$D_x = 8(-1)-2(1) = -10$, so $x = -10/-5 = 2$

$D_y = 3(1)-8(1) = -5$, so $y = -5/-5 = 1$

Solution: $(2, 1)$

💡 Replace the column of the variable you're solving for with the constants column.

Determinant Quiz 🎯

Compute Determinants 🧮

1) $\det \begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix}$ = ?

2) $\det \begin{bmatrix} 6 & 3 \\ 2 & 1 \end{bmatrix}$ = ?

3) Cramer: $\begin{cases} x+y=5 \\ 2x-y=1 \end{cases}$. $D$ = ?

Determinant Concepts 🔽

Exit Quiz ✅

🔄 Inverse Matrices

Part 4 of 7

What Is an Inverse?

If $A^{-1}$ exists, then:

$A \cdot A^{-1} = A^{-1} \cdot A = I$

Like division: $A^{-1}$ "undoes" multiplication by $A$.

2×2 Inverse Formula

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \implies A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$

Steps: swap diagonal, negate off-diagonal, divide by det.

Example

$A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}, \quad \det = 3-2 = 1$

$A^{-1} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}$

Verify: $\begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ✓

📐 Solving Systems with Inverses

The Matrix Equation

$A\vec{x} = \vec{b} \implies \vec{x} = A^{-1}\vec{b}$

Example

$\begin{cases} 3x+y=5 \\ 2x+y=4 \end{cases}$

$A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix}, \quad A^{-1} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix}$

$\vec{x} = \begin{bmatrix} 1 & -1 \\ -2 & 3 \end{bmatrix} \begin{bmatrix} 5 \\ 4 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

Solution: $x = 1, y = 2$.

When Does $A^{-1}$ NOT Exist?

When $\det(A) = 0$ (singular matrix). No unique solution exists.

📋 Properties of Inverses

Key Properties

The "Socks and Shoes" Rule

$(AB)^{-1} = B^{-1}A^{-1}$

Like taking off socks and shoes: reverse order!

Finding Inverse by Row Reduction

$[A | I] \xrightarrow{\text{row ops}} [I | A^{-1}]$

Start with identity augmented, reduce left side to identity → right side becomes the inverse.

Inverse Quiz 🎯

Find the Inverse 🧮

$A = \begin{bmatrix} 4 & 3 \\ 1 & 1 \end{bmatrix}$, $\det = ?$

1) $\det(A)$ = ?

2) $A^{-1}$ top-left entry = ?

3) $A^{-1}$ top-right entry = ?

Inverse Concepts 🔽

Exit Quiz ✅

📊 Row Reduction & Gaussian Elimination

Part 5 of 7

Elementary Row Operations

Three legal moves (they don't change solutions):

Goal: Row Echelon Form (REF)

$\begin{bmatrix} 1 & * & * & | & * \\ 0 & 1 & * & | & * \\ 0 & 0 & 1 & | & * \end{bmatrix}$

Leading 1s form a staircase pattern.

Reduced Row Echelon Form (RREF)

$\begin{bmatrix} 1 & 0 & 0 & | & * \\ 0 & 1 & 0 & | & * \\ 0 & 0 & 1 & | & * \end{bmatrix}$

Solution reads directly from the right side!

📝 Worked Example

Solve: $\begin{cases} x+2y=5 \\ 3x+4y=11 \end{cases}$

$\left[\begin{array}{cc|c} 1 & 2 & 5 \\ 3 & 4 & 11 \end{array}\right]$

Step 1: $R_2 - 3R_1 \to R_2$

$\left[\begin{array}{cc|c} 1 & 2 & 5 \\ 0 & -2 & -4 \end{array}\right]$

Step 2: $-\frac{1}{2}R_2 \to R_2$

$\left[\begin{array}{cc|c} 1 & 2 & 5 \\ 0 & 1 & 2 \end{array}\right]$

Step 3: $R_1 - 2R_2 \to R_1$

$\left[\begin{array}{cc|c} 1 & 0 & 1 \\ 0 & 1 & 2 \end{array}\right]$

Solution: $x = 1, y = 2$ ✓

🔢 3×3 Example

$\begin{cases} x+y+z=6 \\ 2x+3y+z=14 \\ x-y+2z=2 \end{cases}$

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 2 & 3 & 1 & 14 \\ 1 & -1 & 2 & 2 \end{array}\right]$

$R_2-2R_1$, $R_3-R_1$:

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & -1 & 2 \\ 0 & -2 & 1 & -4 \end{array}\right]$

$R_3+2R_2$:

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 6 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & -1 & 0 \end{array}\right]$

Back-substitute: $z=0, y=2, x=4$.

Solution: $(4, 2, 0)$ ✓

Row Reduction Quiz 🎯

Row Operations 🧮

$\left[\begin{array}{cc|c} 2 & 4 & 10 \\ 1 & 3 & 7 \end{array}\right]$

After $\frac{1}{2}R_1$: first row becomes $\begin{bmatrix} 1 & ? & | & ? \end{bmatrix}$

1) New $a_{12}$ = ?

2) New $a_{13}$ (constant) = ?

3) After $R_2 - R_1 \to R_2$: new $a_{22}$ = ?

Gauss Concepts 🔽

Exit Quiz ✅

🔄 Matrix Transformations

Part 6 of 7

Matrices as Transformations

Every $2 \times 2$ matrix defines a linear transformation of the plane.

$T(\vec{v}) = A\vec{v}$

Common Transformation Matrices

Example: Reflect $(3, 2)$ over the $x$-axis

$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \begin{bmatrix} 3 \\ 2 \end{bmatrix} = \begin{bmatrix} 3 \\ -2 \end{bmatrix}$

🔄 Rotation Matrices

General Rotation by Angle $\theta$ (CCW)

$R_\theta = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$

Examples

$90°$: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$

$180°$: $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$

$270°$ (or $-90°$): $\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$

Rotate $(1, 0)$ by $60°$

$\begin{bmatrix} \cos 60° & -\sin 60° \\ \sin 60° & \cos 60° \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1/2 \\ \sqrt{3}/2 \end{bmatrix}$

Key Property

$\det(R_\theta) = \cos^2\theta + \sin^2\theta = 1$

Rotations preserve area!

🔗 Composing Transformations

Sequential Transforms = Matrix Product

Reflect over $x$-axis THEN rotate $90°$:

$T = R_{90°} \cdot M_x = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

This is reflection over $y = x$!

Order Matters (Again!)

$R_{90°} \cdot M_x \neq M_x \cdot R_{90°}$

Just like function composition: apply rightmost first.

Dilation + Rotation

Scale by 2 then rotate $45°$:

$T = R_{45°} \cdot \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} \sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2} \end{bmatrix}$

Transformation Matrices Quiz 🎯

Transform Points 🧮

Reflect $(5, -3)$ over the $y$-axis using $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$:

1) New $x$ = ?

2) New $y$ = ?

3) $\det \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ = ? (rotation matrix det)

Transformation Concepts 🔽

Exit Quiz ✅

🏆 Matrices — Full Synthesis

Part 7 of 7

Master Reference

Invertibility Checklist

A matrix is invertible when:

• $\det(A) \neq 0$
• Row reduces to identity
• $A\vec{x} = \vec{b}$ has a unique solution for every $\vec{b}$
• Columns are linearly independent
• Zero is NOT an eigenvalue

🔄 Method Comparison: Solving Systems

Small Systems (2×2)

• Fastest: Cramer's Rule or inverse formula
• $x = D_x/D$, $y = D_y/D$

Medium Systems (3×3)

• Best: Gaussian elimination
• Systematic, always works, handles special cases

Large Systems

• Standard: RREF (computer-assisted)
• Technology: calculators, MATLAB, Python

When Each Method Fails

💡 Gaussian elimination is the most robust method.

🔗 Linear Algebra Preview

Eigenvalues & Eigenvectors

$A\vec{v} = \lambda \vec{v}$ — special vectors that are only scaled (not rotated).

For $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$:

$\det(A - \lambda I) = 0 \implies (2-\lambda)(3-\lambda) = 0$

Eigenvalues: $\lambda = 2, 3$.

Applications of Matrices

• Computer Graphics: transformations for 3D rendering
• Machine Learning: data processing, neural networks
• Economics: input-output models
• Physics: quantum mechanics states
• Cryptography: encoding/decoding messages

From Precalc to Linear Algebra

Precalc matrices → Linear algebra → Abstract algebra → Modern math!

Synthesis Quiz 🎯

Final Calculations 🧮

1) $\det \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ = ?

2) $\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix}$: top entry = ?

3) Eigenvalues of $\begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$: sum = ?

Matrices Master 🔽

Exit Quiz ✅