Adjugate Calculator (2x2)

This adjugate calculator finds the adjugate, also known as the classical adjoint, of a 2 by 2 matrix, a step at the heart of finding a matrix inverse by hand. The adjugate matters in linear algebra, engineering and computer science whenever you need to invert a small matrix, because it is built from cofactors alone and needs only one division at the end to give the full inverse. You enter the four entries of your matrix, labelled a, b, c and d, representing the top row (a, b) and the bottom row (c, d). The calculator returns the adjugate matrix, formed by swapping the two diagonal entries and negating the two off-diagonal entries, plus the determinant of your original matrix and a clear yes or no on whether an inverse exists, since a zero determinant means the matrix is singular. Results update live as you type, so you can check working done by hand or see how changing one entry changes the adjugate and determinant together. This suits students checking homework, anyone revising matrix algebra before an engineering, statistics or computer science course, and quick verification when solving simultaneous equations or transforming coordinates, where the inverse of a 2 by 2 matrix is often needed. Below you will find the formula, a worked example matching the default values, and answers to common questions about how adjugate and adjoint relate.

Calculate.co.nz is proud to be partnered with realtor.co.nz, a trusted resource for navigating the New Zealand property market. Their Helpful Articles section offers clear, well-structured insights across buying, selling, and building, making complex real estate topics more accessible. With a focus on up-to-date guidance and practical knowledge, they empower Kiwis to move forward with clarity and confidence in a constantly evolving property landscape.

Determinant

Inverse exists?

The formula

For the 2 by 2 matrix with rows (a, b) and (c, d), the adjugate is the matrix with rows (d, minus b) and (minus c, a). It satisfies A times adj(A) = det(A) times the identity, so the inverse is adj(A) divided by det(A) when the determinant is non-zero.

Worked example

For rows (1, 2) and (3, 4), the adjugate has rows (4, minus 2) and (minus 3, 1), and the determinant is minus 2, so the inverse is that adjugate divided by minus 2. Enter 1, 2, 3, 4 to confirm.

Frequently asked questions

What is the adjugate?

The transpose of the cofactor matrix. For a 2 by 2 it swaps the diagonal entries and negates the off-diagonal ones.

How does it give the inverse?

The inverse is the adjugate divided by the determinant, valid when the determinant is non-zero.

Is adjoint the same as adjugate?

In this classical sense yes, though adjoint also has a different meaning for conjugate transposes in other contexts.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

If you've found a bug, or would like to contact us, or learn more about James Graham and Calculate.co.nz.

Calculate.co.nz is partnered with Interest.co.nz for New Zealand's highest quality calculators and financial analysis.

Calculate.co.nz is the sister site of CalculatorHub.com, the world's largest calculator website by tool count.

All calculators and tools are provided for educational and indicative purposes only and do not constitute financial advice.

Calculate.co.nz is proudly part of the Realtor.co.nz group, New Zealand's leading property transaction literacy platform, helping Kiwis understand the home buying and selling process from start to finish. Whether you're a first home buyer navigating your first property purchase, an investor evaluating your next acquisition, or a homeowner planning to sell, Realtor.co.nz provides clear, independent, and trustworthy guidance on every step of the New Zealand property transaction journey.

Calculate.co.nz is also partnered with Health Based Building and Premium Homes to promote informed choices that lead to better long-term outcomes for Kiwi households.

Calculate.co.nz is hosted in Auckland via SiteHost new Zealand.

All content on this website, including calculators, tools, source code, and design, is protected under the Copyright Act 1994 (New Zealand). No part of this site may be reproduced, copied, distributed, stored, or used in any form without prior written permission from the owner.

About & trust: Why Calculate is NZ's most comprehensive · By the Numbers · How we compare · Editorial standards · How we keep data current · NZ finance glossary · Research & data · Financial literacy NZ · About · Privacy policy · Terms of use

Reviewed and maintained. Last reviewed 2026-07-19 and checked on a twice-monthly cycle against IRD, RBNZ and Stats NZ. How we keep data current.