Online Inverse Calculator

Calculator Input

Inverse Type

Number x

Slope a

Constant b

Output y

Row 1, Column 1

Row 1, Column 2

Row 1, Column 3

Row 2, Column 1

Row 2, Column 2

Row 2, Column 3

Row 3, Column 1

Row 3, Column 2

Row 3, Column 3

Integer a

Modulus m

Example Data Table

Inverse Type	Input	Expected Result	Check
Reciprocal	x = 8	0.125	8 × 0.125 = 1
Linear Function	f(x) = 3x + 5, y = 20	x = 5	3(5) + 5 = 20
2×2 Matrix	[4, 7; 2, 6]	[0.6, -0.7; -0.2, 0.4]	det = 10
Modular	a = 17, m = 43	38	17 × 38 ≡ 1 mod 43

Formula Used

Reciprocal Inverse

The reciprocal inverse of a nonzero value is:

x⁻¹ = 1 / x

Linear Function Inverse

For a function written as f(x) = ax + b, solve y = ax + b for x.

f⁻¹(y) = (y - b) / a

2×2 Matrix Inverse

For matrix A = [a, b; c, d], the determinant is:

det(A) = ad - bc

The inverse is:

A⁻¹ = 1 / det(A) × [d, -b; -c, a]

3×3 Matrix Inverse

The calculator expands the determinant, builds cofactors, transposes the cofactor matrix, and divides by the determinant.

A⁻¹ = adj(A) / det(A)

Modular Inverse

A modular inverse solves this congruence:

a × x ≡ 1 mod m

It exists only when gcd(a, m) equals 1.

How to Use This Calculator

Select the inverse type from the dropdown menu.
Enter the required values for that selected method.
Press the calculate button.
Read the result table shown above the form.
Check warnings about zero values, determinants, or coprime conditions.
Use CSV export for spreadsheet records.
Use PDF export for reports, notes, or assignments.

Understanding Inverse Calculations

An inverse reverses a mathematical action. It answers a backward question. When a value is multiplied, the reciprocal can divide it back. When a function sends x to y, the inverse tries to return y to x. When a matrix transforms coordinates, its inverse can undo that transformation. This calculator brings these related ideas into one clear workspace.

Why Inverses Matter

Inverses appear in algebra, geometry, coding, engineering, finance, and data work. They help solve equations. They also help test whether a model can be reversed safely. A matrix inverse can solve systems of linear equations. A modular inverse supports number theory and cryptography exercises. A function inverse helps students understand input and output relationships. A reciprocal supports rates, proportions, and unit changes.

Types Included

The reciprocal mode computes one divided by a nonzero number. The linear function mode handles f(x)=ax+b. It shows the inverse rule and can evaluate the original input for a chosen output. Matrix modes calculate determinants and inverse entries. The 2 by 2 method is compact. The 3 by 3 method uses cofactors and an adjugate matrix. The modular mode finds a number that restores a product to one under a chosen modulus.

Reading The Results

Each result appears above the form after submission. The table lists the main value, formula path, and important checks. Watch determinant and zero warnings carefully. A matrix has no inverse when its determinant equals zero. A reciprocal has no result when the number equals zero. A linear inverse needs a nonzero slope. A modular inverse needs coprime values.

Practical Use

Start with the inverse type that matches your problem. Enter clean numeric values. Keep decimals reasonable. Review the formula section before trusting the answer. Use the example table for quick testing. Export the results when you need a record. CSV files suit spreadsheets. PDF files suit reports. For schoolwork, copy the steps and explain the restrictions. Good inverse work always includes both the answer and the condition that makes it valid.

Accuracy Tips

Use exact integers when possible. Round only at the final stage. Compare by multiplying the inverse back through the original expression. This simple check catches entry mistakes and confirms the calculation before final use.

FAQs

1. What does an inverse mean in maths?

An inverse reverses an operation or transformation. It may mean a reciprocal, inverse function, inverse matrix, or modular inverse, depending on the problem type.

2. Can zero have a reciprocal inverse?

No. Zero has no reciprocal inverse because division by zero is undefined. The calculator shows a warning when zero is entered.

3. When does a linear function have an inverse?

A linear function f(x)=ax+b has an inverse when a is not zero. If a equals zero, the function is constant and cannot be reversed uniquely.

4. Why does determinant matter for matrix inverse?

The determinant shows whether a matrix can be reversed. If the determinant equals zero, the matrix is singular and has no inverse.

5. What is a modular inverse?

A modular inverse is a number that makes a product equal to one under a modulus. It solves a × x ≡ 1 mod m.

6. Why does modular inverse need coprime values?

A modular inverse exists only when a and m share no common divisor except one. This condition allows the equation to be solved uniquely.

7. Can I export my inverse results?

Yes. Use the CSV button for spreadsheet output. Use the PDF button for a printable result table.

8. Are decimal matrix results rounded?

Yes. Results are formatted for readability. For exact study work, keep original fractions or compare results using the check values.