## INTRODUCTION

Learning about tensor transformation (See my tensor page in the learning math menu option) I came into Jakobian Matrix calculation. I studied matrices at Chalmers in Göteborg as a tool to solve equations.

I need to learn Matrix calculus.

# INDEX

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Issue 1

Khan Academy has a great introduction to matrices in www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:mat-intro/a/intro-to-matrices

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## Issue 2

Matrix calculus is what I need most to be able to continue with tensor calculations.

**Number of columns in the first matrix must be equal to the number of raws in the second matrix.**

If it is not, the multiplication can not be done = it is not defined.

The number of columns in the second matrix decides the number of columns in the resulting/ matrix product.

- A
_{23 }x B_{32 }gives AB_{22}

This multiplication can be commuted to

- B
_{32 }x A_{23x }gives AB_{23}

(Again, as said above the number of columns in A decides the number of columns in AB)

You can **not** multiply A_{12} x B_{32} as the number of raws in B is not equal to the number of columns in A.

**This is for me, the most important part.**

- E
_{23}xD_{32}=ED_{22} - A
_{23}W_{13}=AW_{13}

### Multiplying matrix with identity matrix (I)

If you multiply a matrix A with a identity matrix of A you get A

I x A = A x I = A

OBS. Multiplication with identity matrix is commutative if you have square matrices.

### How to calculate A^{-1} the inverted Matrix of A

If A x A^{-1} = I (he identity matrix) then you have found the inverted matrix of A.

To find the inverted matrix of a square 2×2 matrix you have to use this formula

Khan explains in a to difficult way how to invert a square matrix so I choosed this video instead from Professor Dave Explains

This is Khans explanation, for you to compare.

**When matrix multiplication are commutative like a x b = b x a**

- D
_{33}B_{22 }= not defined! (numbers of rows not equal to numbers of columns in B) - Matrix multiplication is therefore not commutative in this case
- A
_{52}B_{23}=AB_{53} - A
_{52}B_{23}= not defined_{(}numbers of rows not equal to numbers of columns in B)

- A

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## conclusion