Tool to calculate matrix products. Matrix product algebra consists of the multiplication of matrices (square or rectangular).
Matrix Product - 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!
$ M_1=[a_{ij}] $ is a matrix of $ m $ lines and $ n $ columns and $ M_2=[b_{ij}] $ is a matrix of $ n $ lines and $ p $ columns (2x2,2x3,3x2,3x3,etc.). The matrix product $ M_1.M_2 = [c_{ij}] $ is a matrix of $ m $ lines and $ p $ columns, with: $$ \forall i, j : c_{ij} = \sum_{k=1}^n a_{ik}b_{kj} $$
Example: $$ \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 \times 1 + 2 \times 0 & 1 \times 0 + 2 \times 1 \\ 3 \times 1 + 4 \times 0 & 3 \times 0 + 4 \times 1 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$
The order of the operands matters with matrix computations, so $$ M_1.M_2 \neq M_2.M_1 $$
The product of the matrix $ M=[a_{ij}] $ by a scalar (number) $ \lambda $ is a matrix of the same size than the initial matrix $ M $, with each items of the matrix multiplied by $ \lambda $.
$$ \lambda M = [ \lambda a_{ij} ] $$
Associativity : $$ A \times (B \times C) = (A \times B) \times C $$
Distributivity : $$ A \times (B + C) = A \times B + A \times C $$
$$ (A + B) \times C = A \times C + B \times C $$
$$ \lambda (A \times B) = (\lambda A) \times B = A \times (\lambda B) $$
There is a matrix product compatible with any matrix sizes: the Kronecker product.
dCode retains ownership of the online 'Matrix Product' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Matrix Product' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Matrix Product' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Matrix Product' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.
Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!
Thanks to your feedback and relevant comments, dCode has developed the best 'Matrix Product' tool, so feel free to write! Thank you!