Tool to invert a matrix. The inverse of a square matrix M is a matrix denoted M^-1 such as que M.M^-1=I where I is the identity matrix.

Inverse of a Matrix - dCode

Tag(s) : Matrix

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!

A suggestion ? a feedback ? a bug ? an idea ? *Write to dCode*!

Tool to invert a matrix. The inverse of a square matrix M is a matrix denoted M^-1 such as que M.M^-1=I where I is the identity matrix.

The inverse of a square matrix is calculated in several ways, the easiest is the cofactor method which necessitate to calculate the determinant of the matrix but also the comatrix and its transposed matrix:

$$ M^{-1}=\frac1{\det M} \,^{\operatorname t}\!{{\rm com} M} = \frac1{\det M} \,^{\rm t}\!C $$

The dCode calculator works for any size of square matrix.

For a 2x2 matrix (order 2):

$$ M^{-1} = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}^{-1} = \frac{1}{\det(M)} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \\ \end{bmatrix} $$

__Example:__ $$ M = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ \end{bmatrix} \Rightarrow M^{-1} = \frac{1}{\det(M)} \begin{bmatrix} 4 & -2 \\ -3 & 1 \\ \end{bmatrix} = -\frac{1}{2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \\ \end{bmatrix} $$

For a 3x3 matrix (order 3):

$$ M^{-1} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}^{-1} = \left( \begin{bmatrix} \frac{e i-f h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c h-b i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{b f-c e}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{f g-d i}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{a i-c g}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{c d-a f}{-c e g+b f g+c d h-a f h-b d i+a e i} \\ \frac{d h-e g}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{b g-a h}{-c e g+b f g+c d h-a f h-b d i+a e i} & \frac{a e-b d}{-c e g+b f g+c d h-a f h-b d i+a e i} \end{bmatrix} \right) $$

It is essential that the determinant of the matrix to be inverted is not equal to zero for the matrix to be invertible.

A matrix is invertible if its determinant is non-zero (different from 0).

A non **invertible matrix** is called *singular* (inversion is not possible).

Avoid the term *inversible* which is wrong.

A matrix with a determinant equal to 0 is not invertible. It does not have an inverse, it is not possible to calculate its inverse.

The multiplication of the matrix by its inverse must give the identity matrix. So the computation of $ M . M^{-1} = I $.

The principle is the same, but instead of calculating the determinant, calculate the modular inverse of the matrix determinant.

dCode retains ownership of the online 'Inverse of a Matrix' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) no data, script or API access will be for free, same for Inverse of a Matrix download for offline use on PC, tablet, iPhone or Android !

Please, check our community Discord for help requests!

- Square Matrix Inverse Calculator NxN
- Matrix Modular Inverse Calculator
- How to calculate the inverse of an invertible matrix?
- How to prove that a matrix is invertible?
- How to inverse a matrix with zero determinant?
- How to check that a matrix is the inverse of another?
- How to calculate the modular inverse of a matrix?

inverse,matrix,square,identity,inversion,invertible,singular

Source : https://www.dcode.fr/matrix-inverse

© 2021 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.

Feedback

▲