04-11-2024 - Analytic Geometry - Matrix Multiplication Example [EN]-[IT]

View this thread on: d.buzz | hive.blog | peakd.com | ecency.com
·@stefano.massari·
0.000 HBD
04-11-2024 - Analytic Geometry - Matrix Multiplication Example [EN]-[IT]
![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23zbaDQGvqq129zxS8SuvcvTS75R8jJno1HLERHKnF1EjMYG4DUCqTUTG6sFhUXYyyMz9.png)




---

*~~~ La versione in italiano inizia subito dopo la versione in inglese ~~~*

---


**ENGLISH**
**04-11-2024 - Analytic Geometry - Matrix Multiplication Example [EN]-[IT]**
With this post I would like to give a brief instruction regarding the topic mentioned in the subject
(code notes: X_086)

***Matrix Multiplication Example***
Let's proceed with the multiplication between the following matrices

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23tw9kFRku8tm4ENnp8YNR2MPSe2gNaUK9DZRavBQnVmvB23FZzRtWMceWgZ7v4V1T3wE.png)

The result for us will be the matrix C which will have the following terms as result
c11​=a11​⋅b11​+a12​⋅b21​
c12​=a11​⋅b12​+a12​⋅b22​
c21​=a21​⋅b11​+a22​⋅b21
c22​=a21​⋅b12​+a22​⋅b22​

Let's replace the terms
c11​=1​⋅1​+1​⋅2
c12​=1​⋅3+1⋅2
c21​=0​⋅1​+1​⋅1
c22​=0​⋅3+1​⋅2

We will obtain the following results
c11​=1​⋅1​+1​⋅2 = 1 + 2 = 3
c12​=1​⋅3+1⋅2 = 3 + 2 = 5
c21​=0​⋅1​+1​⋅1 = 0 + 1 = 1
c22​=0​⋅3+1​⋅2 = 0 + 2 = 2

So the resulting matrix will be the following

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/EoAgtrap6yCYbKY4jjNBgPJGXDkvXZRHkZv9CNiRck141FYvNZBiyFEKwU2xBkqyKEP.png)

**Inverse matrix**
Let's try to find the inverse matrix of

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uFwixvS5YzVqUdBsnc4XJ4ANA43fRn9zSc8SA4GtgnazB3m4xAB8fgCgr1Tno1QixHn.png)

Consider the following system

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23t8D6NyXYKbp5ijhHYReoXDf5jhQwrLbYXTiekweRvC6KN5RErdSaM45XW1wXRR8Ssg3.png)

The unknown is a matrix. Now let's do the two products row by column.

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23t8D6MsqZNRXscyouqR9ZmB2WkxCZcrW5EZTMGCgzC4iZQTACUcxkCYAgE5gMoJXkiSK.png)

At this point we have a system of 8 equations
1 x11 + 1 x21 = 1
1 x12 + 1 x22 = 0
0 x11 + 1 x21 = 0
0 x12 + 1 x22 = 1
1 x11 + 0 x12 = 1
1 x11 + 1 x12 = 0
1 x21 + 0 x22 = 0
1 x21 + 1 x22 = 1

For solution has the quaterna (x11,x12,x21,x22) = (1, -1, 0, 1)

Let's do two checks with the first two rows and the last one
1 x11 + 1 x21 = 1 ---> 1 x 1 + 1 x 0 = 1
1 x12 + 1 x22 = 0 ---> 1 x -1 + 1 x 1 = 0
…
1 x21 + 1 x22 = 1 ---> 1 x 0 + 1 x 1 = 1

***Conclusions***
Matrix multiplications are obtained by multiplying the rows with the columns.

***Question***
In my opinion matrix multiplications are not an easy thing to understand, what do you think?





---

https://images.hive.blog/1536x0/https://files.peakd.com/file/peakd-hive/green77/gGQutTRs-hive-spacer.png

---


**[ITALIAN]**
**04-11-2024 - Geometria analitica - Esempio moltiplicazione di matrici [EN]-[IT]**
Con questo post vorrei dare una breve istruzione a riguardo dell’argomento citato in oggetto
(code notes: X_086)

***Esempio moltiplicazione di matrici***
Procediamo con la moltiplicazione tra le seguenti matrici

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23tw9kFRku8tm4ENnp8YNR2MPSe2gNaUK9DZRavBQnVmvB23FZzRtWMceWgZ7v4V1T3wE.png)

Il risultato per noi sarà la matrice C che avrà i seguenti termini come risultato
c11​=a11​⋅b11​+a12​⋅b21​
c12​=a11​⋅b12​+a12​⋅b22​
c21​=a21​⋅b11​+a22​⋅b21
c22​=a21​⋅b12​+a22​⋅b22​

Andiamo a sostituire i termini
c11​=1​⋅1​+1​⋅2
c12​=1​⋅3+1⋅2
c21​=0​⋅1​+1​⋅1
c22​=0​⋅3+1​⋅2

Otterremo i seguenti risultati
c11​=1​⋅1​+1​⋅2 = 1 + 2 = 3
c12​=1​⋅3+1⋅2 = 3 + 2 = 5
c21​=0​⋅1​+1​⋅1 = 0 + 1 = 1
c22​=0​⋅3+1​⋅2 = 0 + 2 = 2

Quindi la matrice risultante sarà la seguente

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/EoAgtrap6yCYbKY4jjNBgPJGXDkvXZRHkZv9CNiRck141FYvNZBiyFEKwU2xBkqyKEP.png)

**Matrice inversa**
Proviamo a trovare la matrice inversa di 

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23uFwixvS5YzVqUdBsnc4XJ4ANA43fRn9zSc8SA4GtgnazB3m4xAB8fgCgr1Tno1QixHn.png)

Consideriamo il seguente sistema

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23t8D6NyXYKbp5ijhHYReoXDf5jhQwrLbYXTiekweRvC6KN5RErdSaM45XW1wXRR8Ssg3.png)

L’incognita è una matrice. Ora facciamo i due prodotti righe per colonne.

![image.png](https://files.peakd.com/file/peakd-hive/stefano.massari/23t8D6MsqZNRXscyouqR9ZmB2WkxCZcrW5EZTMGCgzC4iZQTACUcxkCYAgE5gMoJXkiSK.png)

A questo punto abbiamo un sistema di 8 equazioni
1 x11 + 1 x21 = 1
1 x12 + 1 x22 = 0
0 x11 + 1 x21 = 0
0 x12 + 1 x22 = 1
1 x11 + 0 x12 = 1
1 x11 + 1 x12 = 0
1 x21 + 0 x22 = 0
1 x21 + 1 x22 = 1

Per soluzione ha la quaterna (x11,x12,x21,x22) = (1, -1, 0, 1)

Facciamo due verifiche con le prime due righe e l’ultima
1 x11 + 1 x21 = 1  --->  1 x 1 + 1 x 0 = 1
1 x12 + 1 x22 = 0 --->  1 x -1 + 1 x 1 = 0
…
1 x21 + 1 x22 = 1 --->  1 x 0 + 1 x 1 = 1

***Conclusioni***
Le moltiplicazioni con le matrici si ottengono moltiplicando le righe con le colonne.

***Domanda***
Secondo me le moltiplicazioni tra matrici non sono una cosa facile da comprendere, voi che ne pensate?

**THE END**
👍 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,