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🎓 Master Mathematics Through Learning

Learn Mathematical Concepts with Expert Guidance

Explore comprehensive tutorials, step-by-step explanations, and interactive examples to master mathematics from basic concepts to advanced topics.

500+
Tutorials
8
Learning Domains
3
Difficulty Levels

Arithmetic Learning Resources

Build your foundational skills with arithmetic tutorials covering basic operations, fractions, percentages, and more. Start with beginner concepts and work your way up.

Beginner #AR-001
⏱️ 15 min

Understanding Order of Operations (PEMDAS)

Learn the correct order for solving mathematical expressions: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Master this fundamental concept to avoid common calculation errors.

Key Insight: Multiplication and division have equal precedence and are evaluated left to right. The same applies to addition and subtraction.

💡 Key Insight

Step-by-Step Solution

Solve: 6 + 4 × 2² ÷ (3 - 1)

1
Parentheses First
(3 - 1) = 2
Always solve expressions inside parentheses first
2
Exponents
2² = 4
Calculate exponents after parentheses
3
Multiplication & Division (Left to Right)
4 × 4 = 16, then 16 ÷ 2 = 8
Multiplication and division have equal priority
4
Addition
6 + 8 = 14
Addition comes last in the order of operations
Final Answer
14
Beginner #AR-002
⏱️ 20 min

Working with Fractions and Percentages

Master fraction operations and percentage calculations with real-world examples. Learn to convert between fractions, decimals, and percentages.

Key Insight: Percent means "per hundred." To convert a percentage to a decimal, divide by 100. To convert a fraction to a percentage, multiply by 100.

💡 Key Insight

Step-by-Step Solution

A shirt costs $80 with a 25% discount. What's the final price?

1
Calculate Discount Amount
25% of 80 = 0.25 × 80 = 20
Convert percentage to decimal: 25% = 0.25
2
Subtract Discount from Original Price
80 - 20 = 60
Subtract the discount amount from original price
3
Alternative Method
80 × (1 - 0.25) = 80 × 0.75 = 60
Direct calculation: Original price × (1 - discount rate)
Final Price
$60
Intermediate #AR-003
⏱️ 25 min

Advanced Fraction Operations

Learn complex fraction operations including addition, subtraction, multiplication, and division of mixed numbers and improper fractions.

Key Insight: When adding or subtracting fractions, find a common denominator. When multiplying, multiply numerators and denominators directly. When dividing, multiply by the reciprocal.

💡 Key Insight

Step-by-Step Solution

Simplify: 3/4 + 2/5 × 10/3 - 1/2

1
Multiplication First (Order of Operations)
2/5 × 10/3 = 20/15 = 4/3
Multiply numerators and denominators, then simplify
2
Rewrite Expression
3/4 + 4/3 - 1/2
Now we have addition and subtraction of fractions
3
Find Common Denominator
3/4 = 9/12, 4/3 = 16/12, 1/2 = 6/12
Lowest common denominator is 12
4
Combine Fractions
9/12 + 16/12 - 6/12 = 19/12
Add and subtract numerators with common denominator
Final Answer
19/12 or 1⁷/₁₂
⌨️

Interactive Learning: Percentage Application

If a population grows from 500 to 650, what is the percentage increase?

Enter your answer as a percentage (without the % sign):

Algebra Learning Resources

Master algebraic concepts through comprehensive tutorials. Solve equations, work with polynomials, and understand functional relationships with our expert guidance.

Beginner #AL-001
⏱️ 20 min

Solving Linear Equations

Learn to solve basic linear equations using inverse operations. Understand the concept of variables and how to isolate them to find solutions.

Key Insight: Whatever you do to one side of the equation, you must do to the other side to maintain equality.

💡 Key Insight

Step-by-Step Solution

Solve: 3x + 7 = 22

1
Subtract 7 from Both Sides
3x + 7 - 7 = 22 - 7
Isolate the term with the variable by removing constants
2
Simplify
3x = 15
Both sides are now simplified
3
Divide Both Sides by 3
x = 15 ÷ 3
Isolate the variable by dividing by its coefficient
Solution
x = 5
Intermediate #AL-002
⏱️ 30 min

Quadratic Equations and Factoring

Master solving quadratic equations using factoring, completing the square, and the quadratic formula. Understand the relationship between factors and roots.

Key Insight: The solutions to a quadratic equation are the x-values where the parabola crosses the x-axis. These are also called roots or zeros.

💡 Key Insight

Step-by-Step Solution

Solve: x² - 5x + 6 = 0

1
Identify Factor Pairs
Factors of 6 that add to -5: -2 and -3
Look for two numbers that multiply to 6 and add to -5
2
Factor the Quadratic
(x - 2)(x - 3) = 0
Write the quadratic as product of binomials
3
Apply Zero Product Property
x - 2 = 0 or x - 3 = 0
If product equals zero, at least one factor must be zero
4
Solve Each Equation
x = 2 or x = 3
Solve the simple linear equations
Solutions
x = 2, 3
Advanced #AL-003
⏱️ 45 min

Systems of Equations

Learn to solve systems of linear equations using substitution, elimination, and matrix methods. Understand when each method is most appropriate.

Key Insight: The solution to a system of equations represents the point(s) where the graphs of the equations intersect.

💡 Key Insight

Step-by-Step Solution

Solve the system: 2x + 3y = 13 and 3x - 2y = 4

1
Multiply Equations to Eliminate y
First equation × 2: 4x + 6y = 26
Second equation × 3: 9x - 6y = 12
Multiply to make coefficients of y opposites
2
Add Equations
(4x + 6y) + (9x - 6y) = 26 + 12
Add equations to eliminate y variable
3
Simplify
13x = 38
y terms cancel out
4
Solve for x
x = 38/13
Divide both sides by 13
5
Substitute to Find y
2(38/13) + 3y = 13 → 76/13 + 3y = 13
3y = 13 - 76/13 = (169-76)/13 = 93/13
y = 93/39 = 31/13
Substitute x value into first equation
Solution
x = 38/13, y = 31/13
ƒ

Interactive Learning: Function Evaluation

Given \( f(x) = 2x^2 - 3x + 5 \), find \( f(4) \).

Enter your answer:

Geometry Learning Resources

Explore geometric concepts through visual tutorials. Learn about shapes, angles, areas, and volumes with interactive examples and practical applications.

Beginner #GE-001
⏱️ 25 min

Basic Geometric Shapes and Properties

Learn about fundamental geometric shapes including triangles, quadrilaterals, circles, and their properties. Understand angles, perimeter, and area calculations.

Key Insight: The sum of angles in any triangle is always 180 degrees. The sum of angles in any quadrilateral is always 360 degrees.

💡 Key Insight

Step-by-Step Solution

Find the area of a triangle with base = 12 cm and height = 8 cm

1
Recall Triangle Area Formula
A = ½ × b × h
Area equals one-half times base times height
2
Substitute Values
A = ½ × 12 × 8
Plug in base = 12 and height = 8
3
Calculate
A = 6 × 8 = 48
First multiply ½ × 12 = 6, then 6 × 8 = 48
Area
48 cm²
Intermediate #GE-002
⏱️ 35 min

Pythagorean Theorem and Applications

Master the Pythagorean theorem and its applications in right triangles. Learn to find missing side lengths and solve real-world problems.

Key Insight: The Pythagorean theorem only applies to right triangles. The square of the hypotenuse equals the sum of squares of the other two sides.

💡 Key Insight

Step-by-Step Solution

In a right triangle, hypotenuse = 13 cm and one leg = 5 cm. Find the other leg.

1
Write Pythagorean Theorem
a² + b² = c²
Where c is hypotenuse, a and b are legs
2
Substitute Known Values
5² + b² = 13²
Let a = 5, c = 13, find b
3
Calculate Squares
25 + b² = 169
5² = 25, 13² = 169
4
Isolate b²
b² = 169 - 25 = 144
Subtract 25 from both sides
5
Take Square Root
b = √144 = 12
Positive root since length is positive
Other Leg
12 cm
Advanced #GE-003
⏱️ 45 min

Circle Theorems and Applications

Explore advanced circle properties including chord theorems, tangent properties, inscribed angles, and cyclic quadrilaterals. Learn to prove geometric relationships.

Key Insight: The angle subtended by an arc at the center is twice the angle subtended at any point on the remaining part of the circle.

💡 Key Insight

Step-by-Step Solution

In a circle, chord AB subtends an angle of 60° at the center. Find the angle it subtends at any point on the major arc.

1
Recall Circle Theorem
Angle at center = 2 × Angle at circumference
For the same arc, central angle is twice inscribed angle
2
Apply Theorem
Angle at circumference = ½ × 60°
Divide central angle by 2
3
Calculate
½ × 60° = 30°
Simple division
Angle at circumference
30°
📐

Interactive Learning: Circle Area

A circle has a circumference of 31.4 units. What is its area? (Use π = 3.14)

Enter your answer (round to one decimal place):

Calculus Learning Resources

Master the mathematics of change through differential and integral calculus. Learn limits, derivatives, integrals, and their applications in real-world problems.

Beginner #CA-001
⏱️ 30 min

Introduction to Limits

Understand the fundamental concept of limits in calculus. Learn to evaluate limits algebraically, graphically, and numerically as a function approaches a specific value.

Key Insight: A limit describes the value that a function approaches as the input approaches some value, not necessarily the function's value at that point.

💡 Key Insight

Step-by-Step Solution

Find: limx→2 (x² - 4)/(x - 2)

1
Direct Substitution
(2² - 4)/(2 - 2) = (4-4)/0 = 0/0
Indeterminate form - need algebraic manipulation
2
Factor Numerator
x² - 4 = (x-2)(x+2)
Difference of squares factorization
3
Simplify
(x-2)(x+2)/(x-2) = x+2 for x ≠ 2
Cancel common factor (x-2)
4
Evaluate Limit
limx→2 (x+2) = 2+2 = 4
Now direct substitution works
Limit
4
Intermediate #CA-002
⏱️ 40 min

Basic Differentiation Rules

Master the power rule, product rule, quotient rule, and chain rule for differentiation. Learn to find derivatives of polynomial, exponential, and trigonometric functions.

Key Insight: The derivative represents the instantaneous rate of change. Geometrically, it's the slope of the tangent line at a point on the curve.

💡 Key Insight

Step-by-Step Solution

Find derivative of f(x) = 3x⁴ - 2x³ + 5x - 7

1
Apply Power Rule to Each Term
d/dx(3x⁴) = 12x³
Power rule: d/dx(xⁿ) = n·xⁿ⁻¹
2
Continue Differentiation
d/dx(-2x³) = -6x²
Constant multiple rule applies
3
Linear Term
d/dx(5x) = 5
Derivative of x is 1, times 5 gives 5
4
Constant Term
d/dx(-7) = 0
Derivative of constant is zero
5
Combine Results
f'(x) = 12x³ - 6x² + 5
Sum of derivatives of each term
Derivative
f'(x) = 12x³ - 6x² + 5
Advanced #CA-003
⏱️ 50 min

Integration Techniques

Learn advanced integration methods including substitution, integration by parts, partial fractions, and trigonometric substitution. Solve definite and indefinite integrals.

Key Insight: Integration is the reverse process of differentiation. The definite integral gives the area under a curve between two points.

💡 Key Insight

Step-by-Step Solution

Evaluate: ∫ x·e²ˣ dx

1
Choose u and dv
Let u = x, dv = e²ˣ dx
For integration by parts: ∫ u dv = uv - ∫ v du
2
Find du and v
du = dx, v = ∫ e²ˣ dx = ½·e²ˣ
Differentiate u, integrate dv
3
Apply Integration by Parts Formula
∫ x·e²ˣ dx = x·(½·e²ˣ) - ∫ (½·e²ˣ) dx
∫ u dv = uv - ∫ v du
4
Simplify
= ½·x·e²ˣ - ½ ∫ e²ˣ dx
Factor out constants
5
Integrate Remaining Term
= ½·x·e²ˣ - ½·(½·e²ˣ) + C
∫ e²ˣ dx = ½·e²ˣ
6
Final Simplification
= ½·e²ˣ·(x - ½) + C = (e²ˣ/4)·(2x - 1) + C
Factor out common terms
Integral
(e²ˣ/4)·(2x - 1) + C
∫

Interactive Learning: Basic Derivative

Find the derivative of \( f(x) = 4x^3 - 3x^2 + 2x - 5 \) at \( x = 1 \).

Enter your answer:

Statistics Learning Resources

Learn to collect, analyze, interpret, and present data. Master descriptive statistics, probability distributions, hypothesis testing, and statistical inference.

Beginner #ST-001
⏱️ 25 min

Descriptive Statistics Basics

Learn to calculate and interpret measures of central tendency (mean, median, mode) and measures of dispersion (range, variance, standard deviation).

Key Insight: The mean is sensitive to outliers, while the median is robust. Standard deviation measures how spread out data points are from the mean.

💡 Key Insight

Step-by-Step Solution

Calculate descriptive statistics for: {4, 8, 6, 5, 3, 8, 9}

1
Calculate Mean
(4+8+6+5+3+8+9)/7 = 43/7 = 6.14
Sum all values, divide by count (n=7)
2
Find Median
Sorted: {3,4,5,6,8,8,9} → Median = 6
Middle value of sorted data (4th of 7 values)
3
Determine Mode
8 appears twice, others once → Mode = 8
Most frequent value
4
Calculate Range
9 - 3 = 6
Maximum minus minimum
Results
Mean=6.14, Median=6, Mode=8, Range=6
Intermediate #ST-002
⏱️ 35 min

Probability Distributions

Understand discrete and continuous probability distributions including binomial, normal, and Poisson distributions. Learn to calculate probabilities and expected values.

Key Insight: The normal distribution is symmetric and bell-shaped. Approximately 68% of data falls within 1 standard deviation of the mean in a normal distribution.

💡 Key Insight

Step-by-Step Solution

Binomial distribution: n=10 trials, p=0.3 success probability. Find probability of exactly 4 successes.

1
Binomial Probability Formula
P(X=k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ
Where C(n,k) is combinations
2
Calculate Combinations
C(10,4) = 10!/(4!×6!) = 210
Number of ways to choose 4 from 10
3
Apply Formula
P(X=4) = 210 × (0.3)⁴ × (0.7)⁶
Substitute values: k=4, n=10, p=0.3
4
Calculate Powers
(0.3)⁴ = 0.0081, (0.7)⁶ = 0.117649
Raise probabilities to appropriate powers
5
Multiply
210 × 0.0081 × 0.117649 = 0.2001
Final multiplication
Probability
0.2001 or 20.01%
Advanced #ST-003
⏱️ 45 min

Hypothesis Testing

Learn to formulate null and alternative hypotheses, calculate test statistics, determine p-values, and make statistical decisions using z-tests and t-tests.

Key Insight: The p-value represents the probability of observing results as extreme as those in your sample, assuming the null hypothesis is true.

💡 Key Insight

Step-by-Step Solution

Test if mean > 100 with sample: n=25, x̄=105, σ=15, α=0.05

1
State Hypotheses
H₀: μ = 100, H₁: μ > 100
Null hypothesis (no effect) vs alternative (one-tailed)
2
Calculate Test Statistic
z = (105-100)/(15/√25) = 5/3 = 1.67
z = (x̄ - μ)/(σ/√n)
3
Find Critical Value
z-critical (one-tailed, α=0.05) = 1.645
From z-table for right-tailed test at 5% significance
4
Compare and Decide
1.67 > 1.645 → Reject H₀
Test statistic exceeds critical value
5
Find p-value
P(Z > 1.67) = 0.0475
Probability of z > 1.67 from standard normal table
Conclusion
Reject null hypothesis at 5% significance level
📈

Interactive Learning: Standard Deviation

Calculate the standard deviation for this data set: {2, 4, 6, 8, 10}

Enter your answer (round to two decimal places):

Number Theory Learning Resources

Explore properties of integers, prime numbers, divisibility, modular arithmetic, and Diophantine equations. Discover the fascinating world of pure mathematics.

Beginner #NT-001
⏱️ 20 min

Divisibility Rules and Prime Numbers

Learn divisibility tests for numbers 2 through 11. Understand prime numbers, composite numbers, and the Fundamental Theorem of Arithmetic.

Key Insight: A prime number has exactly two distinct positive divisors: 1 and itself. Every integer greater than 1 is either prime or can be uniquely factored into primes.

💡 Key Insight

Step-by-Step Solution

Find prime factorization of 84

1
Divide by Smallest Prime
84 ÷ 2 = 42
84 is even, so divisible by 2
2
Continue Dividing by 2
42 ÷ 2 = 21
42 is still even
3
Divide by Next Prime
21 ÷ 3 = 7
Sum of digits 2+1=3, divisible by 3
4
Identify Prime Factor
7 is prime
7 has no divisors other than 1 and itself
Prime Factorization
84 = 2² × 3 × 7
Intermediate #NT-002
⏱️ 30 min

Greatest Common Divisor and Least Common Multiple

Master algorithms for finding GCD and LCM including prime factorization and the Euclidean algorithm. Learn applications to fraction simplification.

Key Insight: For any two numbers a and b: GCD(a,b) × LCM(a,b) = a × b. The Euclidean algorithm is efficient for finding GCD of large numbers.

💡 Key Insight

Step-by-Step Solution

Find GCD and LCM of 48 and 180 using Euclidean algorithm

1
First Division
180 ÷ 48 = 3 remainder 36
Divide larger by smaller number
2
Second Division
48 ÷ 36 = 1 remainder 12
Divide previous divisor by remainder
3
Third Division
36 ÷ 12 = 3 remainder 0
Continue until remainder is 0
4
Identify GCD
GCD = 12 (last non-zero remainder)
GCD is the last non-zero remainder
5
Calculate LCM
LCM = (48 × 180) ÷ 12 = 720
Using formula: GCD × LCM = a × b
Results
GCD = 12, LCM = 720
Advanced #NT-003
⏱️ 40 min

Modular Arithmetic and Congruences

Explore modular arithmetic, congruence relations, and their properties. Learn to solve linear congruences and understand applications in cryptography.

Key Insight: Two numbers are congruent modulo n if they have the same remainder when divided by n. Modular arithmetic is cyclic with period n.

💡 Key Insight

Step-by-Step Solution

Solve: 7x ≡ 3 (mod 11)

1
Find Modular Inverse of 7 mod 11
7 × 8 = 56 ≡ 1 mod 11
56 ÷ 11 = 5 remainder 1
2
Identify Inverse
Inverse of 7 is 8 mod 11
Since 7×8 ≡ 1 mod 11
3
Multiply Both Sides by Inverse
x ≡ 3 × 8 (mod 11)
Multiply congruence by modular inverse
4
Calculate
x ≡ 24 (mod 11)
3 × 8 = 24
5
Simplify Mod 11
24 ≡ 2 (mod 11)
24 ÷ 11 = 2 remainder 2
Solution
x ≡ 2 mod 11
🔢

Interactive Learning: Modular Arithmetic

Calculate: \( 3^{10} \mod 7 \)

Hint: Use properties of modular arithmetic

Enter your answer:

Trigonometry Learning Resources

Master trigonometric functions, identities, equations, and their applications. Learn to solve triangles, work with periodic functions, and apply trigonometry to real-world problems.

Beginner #TR-001
⏱️ 25 min

Right Triangle Trigonometry

Learn the six trigonometric ratios (sine, cosine, tangent, and their reciprocals) in right triangles. Understand SOH-CAH-TOA and solve for missing sides and angles.

Key Insight: In a right triangle, sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent. These ratios depend only on the angle, not the triangle size.

💡 Key Insight

Step-by-Step Solution

In a right triangle, angle A = 30°, hypotenuse = 10. Find opposite side.

1
Identify Trigonometric Ratio
sin(30°) = opposite/hypotenuse
Sine relates opposite side to hypotenuse
2
Recall sin(30°) Value
sin(30°) = 0.5
Standard trigonometric value
3
Set Up Equation
0.5 = opposite/10
Substitute known values
4
Solve for Opposite
opposite = 10 × 0.5 = 5
Multiply both sides by 10
Opposite Side
5 units
Intermediate #TR-002
⏱️ 35 min

Trigonometric Identities and Equations

Master fundamental trigonometric identities including Pythagorean, reciprocal, and co-function identities. Learn to prove identities and solve trigonometric equations.

Key Insight: The Pythagorean identity sin²θ + cos²θ = 1 is fundamental. All other trigonometric identities can be derived from basic definitions and this identity.

💡 Key Insight

Step-by-Step Solution

Prove: 1/cosθ - cosθ = sinθ·tanθ

1
Simplify Left Side
1/cosθ - cosθ = (1 - cos²θ)/cosθ
Combine terms with common denominator
2
Apply Pythagorean Identity
1 - cos²θ = sin²θ
From sin²θ + cos²θ = 1
3
Substitute
Left side becomes: sin²θ/cosθ
Replace numerator with sin²θ
4
Simplify Right Side
sinθ·tanθ = sinθ·(sinθ/cosθ) = sin²θ/cosθ
tanθ = sinθ/cosθ
5
Compare Both Sides
Both sides equal sin²θ/cosθ
Identity is proven
Proof Complete
Identity proven ✓
Advanced #TR-003
⏱️ 45 min

Law of Sines and Law of Cosines

Learn to solve oblique (non-right) triangles using the Law of Sines and Law of Cosines. Understand when to use each law and handle ambiguous cases.

Key Insight: Use Law of Sines when you know two angles and one side (AAS or ASA) or two sides and an angle opposite one of them (SSA). Use Law of Cosines for SAS or SSS cases.

💡 Key Insight

Step-by-Step Solution

Triangle: a=7, b=5, C=40°. Find side c.

1
Identify Case
SAS case (two sides and included angle)
Use Law of Cosines
3
Substitute Values
c² = 7² + 5² - 2×7×5×cos40°
Plug in a=7, b=5, C=40°
4
Calculate Squares
c² = 49 + 25 - 70×0.7660
7²=49, 5²=25, cos40°≈0.7660
5
Multiply
c² = 74 - 53.62 = 20.38
70×0.7660=53.62
6
Take Square Root
c = √20.38 ≈ 4.51
Final calculation
Side c
≈ 4.51 units
θ

Interactive Learning: Trig Function Value

Find the exact value of \( \cos(45°) \)

Enter your answer as a fraction or decimal:

Linear Algebra Learning Resources

Master vectors, matrices, linear transformations, and systems of linear equations. Learn concepts essential for computer graphics, machine learning, and engineering applications.

Beginner #LA-001
⏱️ 30 min

Vectors and Vector Operations

Learn vector representation, magnitude, direction, and basic operations including addition, subtraction, scalar multiplication, and dot product.

Key Insight: Vectors have both magnitude and direction. The dot product of two vectors gives a scalar that measures their alignment and can be used to find angles between vectors.

💡 Key Insight

Step-by-Step Solution

Given vectors: u = ⟨2, 3⟩, v = ⟨-1, 4⟩

1
Vector Addition
u + v = ⟨2+(-1), 3+4⟩ = ⟨1, 7⟩
Add corresponding components
2
Dot Product
u · v = (2×-1) + (3×4) = -2 + 12 = 10
Multiply corresponding components and sum
3
Magnitude of u
|u| = √(2² + 3²) = √(4 + 9) = √13 ≈ 3.61
Square root of sum of squared components
Results
Sum = ⟨1,7⟩, Dot = 10, |u| ≈ 3.61
Intermediate #LA-002
⏱️ 40 min

Matrix Operations and Properties

Master matrix addition, multiplication, transposition, and special matrices. Learn about identity matrices, inverses, and determinants.

Key Insight: Matrix multiplication is not commutative (AB ≠ BA in general). The determinant of a matrix indicates whether it has an inverse and relates to the scaling factor of the linear transformation.

💡 Key Insight

Step-by-Step Solution

Multiply matrices: A = [[1,2],[3,4]], B = [[5,6],[7,8]]

1
Element (1,1)
1×5 + 2×7 = 5 + 14 = 19
Row 1 of A × Column 1 of B
2
Element (1,2)
1×6 + 2×8 = 6 + 16 = 22
Row 1 of A × Column 2 of B
3
Element (2,1)
3×5 + 4×7 = 15 + 28 = 43
Row 2 of A × Column 1 of B
4
Element (2,2)
3×6 + 4×8 = 18 + 32 = 50
Row 2 of A × Column 2 of B
Product Matrix
AB = [[19,22],[43,50]]
Advanced #LA-003
⏱️ 50 min

Eigenvalues and Eigenvectors

Understand eigenvalues and eigenvectors, their geometric interpretation, and applications. Learn to find eigenvalues and eigenvectors for 2×2 and 3×3 matrices.

Key Insight: An eigenvector of a matrix is a non-zero vector that only changes by a scalar factor (the eigenvalue) when that matrix is applied to it. They reveal the fundamental directions of the linear transformation.

💡 Key Insight

Step-by-Step Solution

Find eigenvalues of A = [[4,1],[2,3]]

1
Characteristic Equation
det(A - λI) = 0
Set up characteristic equation
2
Construct A - λI
[[4-λ, 1], [2, 3-λ]]
Subtract λ from diagonal elements
3
Calculate Determinant
(4-λ)(3-λ) - (1)(2) = 0
ad - bc formula for 2×2 determinant
4
Expand
λ² - 7λ + 12 - 2 = λ² - 7λ + 10 = 0
Simplify the equation
5
Factor
(λ-2)(λ-5) = 0
Find roots of quadratic
Eigenvalues
λ = 2, 5
🧮

Interactive Learning: Matrix Determinant

Find the determinant of: [[3,1],[4,2]]

Enter your answer:

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