The complete mathematics formulary

The properties of fractions

Elementary operations

Multiplication

Multiplying fractions together is the same as multiplying all the numerators (resp. all the denominators) together.

Division

Dividing by a fraction is the same as multiplying by its inverse.

Addition / substraction

Adding (resp. subtracting) fractions together requires putting them under a common denominator.

Properties

Cross product

Ratio between respectives numerators and denominators

The same ratios are possibles replacing all \( (+) \) by \( (-) \).

Ratio between respectives sums an differences

Addition of numerators and denominators

The same relation is possible replacing \( (+) \) by \( (-) \).

1. Generalization

   On the whole, with a serie of \(n \) numerators and \(m \) denominators:

   $$ \forall F \in \mathbb{Q}, \enspace \forall (a, c, e ...) \in \hspace{0.04em} \mathbb{N}^n, \enspace (b, d, f...) \in \hspace{0.04em} \mathbb{N}^m, \enspace \ \Bigl \{ (b \textcolor{rgb(118 139 240)}{\pm} d \textcolor{rgb(93 183 129)}{\pm} f \textcolor{rgb(232 124 124)}{\pm} ...) \Bigr \} \ \neq 0, $$
   $$ F = \frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \ ... \ \Longrightarrow F = \frac{a \textcolor{rgb(118 139 240)}{\pm} c \textcolor{rgb(93 183 129)}{\pm} e \textcolor{rgb(232 124 124)}{\pm} \ ...}{b \textcolor{rgb(118 139 240)}{\pm} d \textcolor{rgb(93 183 129)}{\pm} f \textcolor{rgb(232 124 124)}{\pm} \ ...}$$
   $$ (\text{with the same-colour signs being the same}) $$

The properties of the powers of x (for natural exponents)

Let \( n\in \mathbb{N}\) be natural number and \( x \in \mathbb{R}\) a real number.

We call \(x^n\) a number \(x\) multiplied \(n\) times by itself:

Product/quotient of powers (having the same base)

Number to the power of zero

Inverse of a power

Power of a product/quotient

Power of a power

The geometrical identity

We call geometrical identity, or Bernoulli's formula , the following expression:

The Newton's binomial \(: (a + b)^n \)

Newton's binomial tells us that :

We can use the Pascal's triangle to find binomials coefficients \(\binom{n}{p}\).

The properties of the binomial\(: \binom{n}{p}\)

Let \((p,n) \in \hspace{0.04em}\mathbb{N}^2 \) be two natural numbers with \( p \leqslant n \).

We call \(\binom{n}{p} \) ("\( p \) among \( n \)") to number of ways to take \( p \) elements among a set of \( n \) elements.

We also call it the binomial, and meets the following definition:

"0 among n" / "n among n"

"1 among n"

Symmetry

The pawn's formula

The Pascal's formula

Horizontal sum from 0 to n

Vertical sum from r to n

Combinatorial analysis and enumerating formulas

All the formulas that follow will always have two cases:

• without repetition
• with repetition : in this case there will be a bar above to mean "with repetition"

  For example, if we note the arrangements without repetition \(A_n\), we will note \(\overline{A_n}\) those with possible repetition.

Permutations

The number of permutations of the elements of a set (without repetition)

For any set \(E\) of \(n\) elements, the number of possible permutations without repetition is:

The number of permutations of the elements of a set (with repetition)

For any set \(E = \{e_1, e_2, e_3, \ ..., \ e_n \}\) with \(k_1, k_2, k_3, ...,k_{n}\) the number of occurrences of each element, the number of possible permutations is:

Arrangements (with a certain order)

The number of arrangements of the elements of a set (without repetition)

The number of arrangements without repetition of \(p\) elements taken from a set of \(n\) are worth:

The number of arrangements of the elements of a set (with repetition)

The number of arrangements with repetition of \(p\) elements taken from a set of \(n\) are worth:

Combinations (without any order)

The number of ways to take distinct elements from a set (without repetition)

The number of ways to take \(p\) elements (distinct and without repetition) in a set of \(n\) elements are worth:

(\(\Longrightarrow\) voir les propriétés du binôme )

The number of ways to take distinct elements from a set (with repetition)

The number of ways to take \(p\) elements (distinct and with repetition) in a set of \(n\) elements are worth:

The number of possible parts of a set

The number of possible parts of a set \(E = \{e_1, e_2, e_3, \ ..., e_n\}\), that's to say :

is worth :

The properties of matrices

For what follows, it is important to establish the following definitions:

1. Operations on matrices
   1. Matrices addition

      Let \((A,B) \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})^2\) be two matrices of the same size.

      $$ \forall (i, j) \in [\![1, n]\!] \times [\![1, p]\!],$$
      $$(A + B)_{i,j} = a_{i,j} + b_{i,j} $$

      In other words, we add each element of the left matrix with the element located at the same position of the right one:

      $$ A + B = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} + \begin{pmatrix} b_{1,1} & b_{1,2} & b_{1,3} & \dots & b_{1, p} \\ b_{2,1} & b_{2,2} & b_{2,3} & \dots & b_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ b_{n,1} & b_{n,2} & b_{n,3} & \dots & b_{n, p} \end{pmatrix} $$
      $$ A + B = \begin{pmatrix} a_{1,1} + b_{1,1} & a_{1,2} + b_{1,2} & a_{1,3} + b_{1,3} & \dots & a_{1, p} + b_{1, p} \\ a_{2,1} + b_{2,1} & a_{2,2} + b_{2,2} & a_{2,3} + b_{2,3} & \dots & a_{2, p} + b_{2,p} \\ \hspace{2em} \vdots & \hspace{2em} \vdots & \hspace{2em} \vdots & \ddots & \hspace{2em} \vdots \\ a_{n,1} + b_{n,1} & a_{n,2} + b_{n,2} & a_{n,3} + b_{n,3} & \dots & a_{n, p} + b_{n,p} \end{pmatrix} $$
   2. Matrices product

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) and \(B \in \hspace{0.03em} \mathcal{M}_{p,q} (\mathbb{K})\) be two matrices.

      To multiply two matrices, we need the left matrix to have the same number of columns as the number of rows of the right one (here \(p\)). As a result, we obtain a matrix \(AB \in \hspace{0.03em} \mathcal{M}_{n,q} (\mathbb{K})\), so having \(n\) lines and \(q\) columns.

      $$ \forall (i, j) \in [\![1, n]\!] \times [\![1, q]\!],$$
      $$(A \times B)_{i,j} = \sum_{k = 1}^p a_{i,k} \times b_{k,j} $$

      For example:

      $$ A \times B = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} \times \begin{pmatrix} b_{1,1} & b_{1,2} & b_{1,3} & \dots & b_{1, q} \\ b_{2,1} & b_{2,2} & b_{2,3} & \dots & b_{2, q} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ b_{p,1} & b_{p,2} & b_{p,3} & \dots & b_{p, q} \end{pmatrix} $$
      $$ A \times B = \begin{pmatrix} \Bigl[a_{1,1} b_{1,1} + a_{1,2} b_{2,1} \ + \ ... \ + \ a_{1,p} b_{p,1} \Bigr] & \Bigl[a_{1,1} b_{1,2} + a_{1,2} b_{2,2} \ + \ ... \ + \ a_{1,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{1,1} b_{1,q} + a_{1,2} b_{2,q} \ + \ ... \ + \ a_{1,p} b_{p,q}\Bigr] \\ \Bigl[a_{2,1} b_{1,1} + a_{2,2} b_{2,1} \ + \ ... \ + \ a_{2,p} b_{p,1}\Bigr] & \Bigl[a_{2,1} b_{1,2} + a_{2,2} b_{2,2} \ + \ ... \ + \ a_{2,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{2,1} b_{1,q} + a_{2,2} b_{2,q} \ + \ ... \ + \ a_{2,p} b_{p,q}\Bigr] \\ \hspace{8em} \vdots & \hspace{8em} \vdots & \hspace{1em} \ddots & \hspace{8em} \vdots \\ \hspace{8em} \vdots & \hspace{8em} \vdots & \hspace{1em} \ddots & \hspace{8em} \vdots \\ \Bigl[a_{n,1} b_{1,1} + a_{n,2} b_{2,1} \ + \ ... \ + \ a_{n,p} b_{p,1}\Bigr] & \Bigl[a_{n,1} b_{1,2} + a_{2,2} b_{2,2} \ + \ ... \ + \ a_{n,p} b_{p,2}\Bigr] & \hspace{1em} \dots \dots \dots \hspace{1em} & \Bigl[a_{n,1} b_{1,q} + a_{n,2} b_{2,q} \ + \ ... \ + \ a_{n,p} b_{p,q}\Bigr] \end{pmatrix} $$

      Be careful, in a general way the matrices product does not have commutative law: \( (A \times B) \neq (B \times A) \) .

   3. Multiplication of a matrix by a scalar \(\lambda\)

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) be a matrix.

      When a matrix is multiplied by a scalar, it affects all its elements.

      $$ \forall (i, j) \in [\![1, n]\!] \times [\![1, p]\!],$$
      $$(\lambda A)_{i,j} = \lambda \ a_{i,j} $$

      For example:

      $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} $$
      $$ \lambda A = \begin{pmatrix} \lambda \ a_{1,1} & \lambda \ a_{1,2} & \lambda \ a_{1,3} & \dots & \lambda \ a_{1, p} \\ \lambda \ a_{2,1} & \lambda \ a_{2,2} & \lambda \ a_{2,3} & \dots & \lambda \ a_{2, p} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \vdots \\ \lambda \ a_{n,1} & \lambda \ a_{n,2} & \lambda \ a_{n,3} & \dots & \lambda \ a_{n, p} \end{pmatrix} $$
   4. Linear combination of matrices

      Let \((A,B) \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})^2\) be two matrices of the same size and \((\lambda, \mu) \in \hspace{0.04em} \mathbb{R}^2\).

      With the previous properties of addition and multiplication by a scalar , we can create linear combinations and:

      $$(\lambda A + \mu B)_{i,j} = \lambda \ a_{i,j} + \mu \ b_{i,j} $$
   5. Matrix transposition

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

      Matrix transposition consists in reverse lines and columns indices for each elements. We note \(A^T\) (sometimes \(^t A\)) the transposed of the matrix \(A\).

      $$ \forall (i, j) \in [\![1, n]\!]^2,$$
      $$ \left(A^T \right)_{i,j} \hspace{0.03em} = a_{j,i} $$

      For example:

      $$ A = \begin{pmatrix} a_{1,1} & \textcolor{rgb(232 124 124)}{a_{1,2}} & \textcolor{rgb(232 124 124)}{a_{1,3}} & \textcolor{rgb(232 124 124)}{\dots} & \textcolor{rgb(232 124 124)}{a_{1, n}} \\ \textcolor{rgb(93 183 129)}{a_{2,1}} & a_{2,2} & \textcolor{rgb(232 124 124)}{a_{2,3}} & \textcolor{rgb(232 124 124)}{\dots} & \textcolor{rgb(232 124 124)}{a_{2, n}} \\ \textcolor{rgb(93 183 129)}{a_{3,1}} & \textcolor{rgb(93 183 129)}{a_{3,2}} & a_{3,3} & \textcolor{rgb(232 124 124)}{\dots} & \textcolor{rgb(232 124 124)}{a_{3, n}} \\ \hspace{1em} \textcolor{rgb(93 183 129)}{\vdots} & \hspace{1em} \textcolor{rgb(93 183 129)}{\vdots} & \hspace{1em} \textcolor{rgb(93 183 129)}{\vdots} & \ddots & \hspace{1em} \textcolor{rgb(232 124 124)}{\vdots} \\ \textcolor{rgb(93 183 129)}{a_{n,1}} & \textcolor{rgb(93 183 129)}{a_{n,2}} & \textcolor{rgb(93 183 129)}{a_{n,3}} & \textcolor{rgb(93 183 129)}{\dots} & a_{n, n} \\ \end{pmatrix} $$

      So, its transposed is:

      $$ A^T = \begin{pmatrix} a_{1,1} & \textcolor{rgb(93 183 129)}{a_{2,1}} & \textcolor{rgb(93 183 129)}{a_{3,1}} & \textcolor{rgb(93 183 129)}{\dots} & \textcolor{rgb(93 183 129)}{a_{n, 1}} \\ \textcolor{rgb(232 124 124)}{a_{1,2}} & a_{2,2} & \textcolor{rgb(93 183 129)}{a_{3,2}} & \textcolor{rgb(93 183 129)}{\dots} & \textcolor{rgb(93 183 129)}{a_{n, 2}} \\ \textcolor{rgb(232 124 124)}{a_{1,3}} & \textcolor{rgb(232 124 124)}{a_{2,3}} & a_{3,3} & \textcolor{rgb(93 183 129)}{\dots} & \textcolor{rgb(93 183 129)}{a_{n, 3}} \\ \hspace{0.8em} \textcolor{rgb(232 124 124)}{\vdots} & \hspace{0.8em} \textcolor{rgb(232 124 124)}{\vdots} & \hspace{0.8em} \textcolor{rgb(232 124 124)}{\vdots} & \ddots & \hspace{0.8em} \textcolor{rgb(93 183 129)}{\vdots} \\ \textcolor{rgb(232 124 124)}{a_{1,n}} & \textcolor{rgb(232 124 124)}{a_{2,n}} & \textcolor{rgb(232 124 124)}{a_{3,n}} & \textcolor{rgb(232 124 124)}{\dots} & a_{n, n} \\ \end{pmatrix} $$

      Only the diagonal remains intact, because when \(i = j\), then \(a_{i,j} = a_{j,i}\).

   6. Inversion of a matrix

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K})\) be a matrix.

      The inverse of the matrix \(A\) is the matrix written \(A^{-1}\) and the same size, such as: \(A A^{-1} = I_n\).

      
      

      To check if a matrix can be inverted, we do have to compute its determinant, and:

      $$ A \text{ is inversible } \Longleftrightarrow det(A) \neq 0 $$
   7. Comatrix

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

      The comatrix of the matrix \(A\) is the matrix noted \(com(A)\), such as:

      $$ \forall (i, j) \in [\![1, n]\!]^2,$$
      $$ com(A)_{i,j} \hspace{0.03em} = C_{i,j} $$
      $$ \text{où } \ \left \{ \begin{gather*} C_{i,j} : \text{cofactors of the element } a_{i, j} \\ M_{i, j} : \text{minor of the element } a_{i, j} \end{gather*} \right \} $$

      \(C_{i, j}\) : cofactors of the element \(a_{i, j}\)

      $$ C_{i,j} = (-1)^{i + j} \times det(M_{i, j}) $$

      \(M_{i, j}\) : minor of the element \(a_{i, j}\)

      The minor of \(a_{i, j}\) is the undermatrix of \(A\) without the line \(i\) and the column \(j\).

      For example, starting from the following matrix \(A\), the minor \(\textcolor{rgb(118 139 240)}{M_{1,1}}\) appears in blue:

      $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} & \dots & a_{1, p} \\ a_{2,1} & \textcolor{rgb(118 139 240)}{a_{2,2}} & \textcolor{rgb(118 139 240)}{a_{2,3}} & \dots & \textcolor{rgb(118 139 240)}{a_{2, p}} \\ a_{3,1} & \textcolor{rgb(118 139 240)}{a_{3,2}} & \textcolor{rgb(118 139 240)}{a_{3,3}} & \dots & \textcolor{rgb(118 139 240)}{a_{3, p}} \\ \hspace{0.5em} \vdots & \hspace{0.5em} \textcolor{rgb(118 139 240)}{\vdots} & \hspace{0.5em} \textcolor{rgb(118 139 240)}{\vdots} & \textcolor{rgb(118 139 240)}{\ddots} & \hspace{0.5em} \textcolor{rgb(118 139 240)}{\vdots} \\ a_{n,1} & \textcolor{rgb(118 139 240)}{a_{n,2}} & \textcolor{rgb(118 139 240)}{a_{n,3}} & \dots & \textcolor{rgb(118 139 240)}{a_{n, p}} \end{pmatrix} $$

      So,

      $$ \textcolor{rgb(118 139 240)}{ M_{1,1} = \begin{pmatrix} a_{2,2} & a_{2,3} & \dots & a_{2, p} \\ a_{3,2} & a_{3,3} & \dots & a_{3, p} \\ \vdots & \hspace{0.5em} \vdots & \ddots & \hspace{0.5em} \textcolor{rgb(118 139 240)}{\vdots} \\ a_{n,2} & a_{n,3} & \dots & a_{n, p} \end{pmatrix} } $$

      For example, starting from the following matrix \(A\):

      $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{pmatrix} $$

      Its comatrix is worth:

      $$ com(A) = \begin{pmatrix} \textcolor{rgb(93 183 129)}{+}\begin{vmatrix} a_{2,2} & a_{2,3} \\ a_{3,2} & a_{3,3} \end{vmatrix} & \textcolor{rgb(232 124 124)}{-}\begin{vmatrix} a_{2,1} & a_{2,3} \\ a_{3,1} & a_{3,3} \end{vmatrix} & \textcolor{rgb(93 183 129)}{+}\begin{vmatrix} a_{2,1} & a_{2,2} \\ a_{3,1} & a_{3,2} \end{vmatrix} \\ \textcolor{rgb(232 124 124)}{-}\begin{vmatrix} a_{1,2} & a_{1,3} \\ a_{3,2} & a_{3,3} \end{vmatrix} & \textcolor{rgb(93 183 129)}{+}\begin{vmatrix} a_{1,1} & a_{1,3} \\ a_{3,1} & a_{3,3} \end{vmatrix} & \textcolor{rgb(232 124 124)}{-}\begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{3,1} & a_{3,2} \end{vmatrix} \\ \textcolor{rgb(93 183 129)}{+}\begin{vmatrix} a_{1,2} & a_{1,3} \\ a_{2,2} & a_{2,3} \end{vmatrix} & \textcolor{rgb(232 124 124)}{-}\begin{vmatrix} a_{1,1} & a_{1,3} \\ a_{2,1} & a_{2,3} \end{vmatrix} & \textcolor{rgb(73 174 65)}{+}\begin{vmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{vmatrix} \end{pmatrix} $$
   8. Matricial writing of a system of linear equations

      A system of linear equations \( (S)\), where the unknown are the variables \(x_{i,j}\), can be written as a product matrix system :

      $$ (S) \enspace \left \{ \begin{gather*} a_1 x_{1,1} + a_2 x_{1,2} + a_3 x_{1,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{1,p} = b_1 \\ a_1 x_{2,1} + a_2 x_{2,2} + a_3 x_{2,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{2,p} = b_2 \\ ........................ ............. \ = \ ..\\ a_1 x_{n,1} + a_2 x_{n,2} + a_3 x_{n,3} + \hspace{0.1em}... \hspace{0.1em}+ a_n x_{n,p} = b_n \\ \end{gather*} \right \} $$
      $$ \Longleftrightarrow$$
      $$ \underbrace{ \begin{pmatrix} x_{1,1} & x_{1,2} & x_{1,3} & \dots & x_{1, p} \\ x_{2,1} & x_{2,2} & x_{2,3} & \dots & x_{2, p} \\ \hspace{0.8em} \vdots & \hspace{0.8em} \vdots & \hspace{0.8em} \vdots & \ddots & \hspace{0.8em} \vdots \\ x_{n,1} & x_{n,2} & x_{n,3} & \dots & x_{n, p} \\ \end{pmatrix} } _\text{X} \times \underbrace{ \begin{pmatrix} a_1 \\ a_2 \\ \hspace{0.3em}\vdots \\ a_n \end{pmatrix} } _\text{A} = \underbrace{ \begin{pmatrix} b_1 \\ b_2 \\ \hspace{0.3em}\vdots \\ b_n \end{pmatrix} } _\text{B} \ \Longleftrightarrow \ XA = B, \ \text{with } \left \{ \begin{gather*} X \in \hspace{0.03em} \mathcal{M}_{n,p} (\mathbb{K}) \\ A \in \hspace{0.03em} \mathcal{M}_{1,p} (\mathbb{K}) \\ B \in \hspace{0.03em} \mathcal{M}_{1,p} (\mathbb{K}) \end{gather*} \right \} $$
   9. Trace of a matrix

      Let \(A \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})\) be a squared matrix of size \(n\).

      We call the trace of a matrix, the sum of all its diagonal elements:

      $$ A = \begin{pmatrix} \textcolor{rgb(118 139 240)}{a_{1,1}} & a_{1,2} & a_{1,3} & \dots & a_{1,n} \\ a_{2,1} & \textcolor{rgb(118 139 240)}{a_{2,2}} & a_{2,3} & \dots & a_{2,n} \\ a_{3,1} & a_{3,2} & \textcolor{rgb(118 139 240)}{a_{3,3}} & \dots & a_{3,n} \\ \hspace{0.1em}\vdots & \hspace{0.1em} \vdots & \hspace{0.1em} \vdots & \textcolor{rgb(118 139 240)}{\ddots} & \hspace{0.1em} \vdots \\ a_{n,1} & a_{n,2} & a_{n,3} & \dots & \textcolor{rgb(118 139 240)}{a_{n,n}} \end{pmatrix} $$
      $$Tr(A) = \sum_{k = 1}^n a_{k,k} = a_{1,1} + a_{2,2} \ + \ ... \ + a_{n,n}$$
2. Specific matrices
   1. Diagonal matrix

      A diagonal is a squared matrix where all the elements are \(0\) except on the main diagonal:

      $$ D_n = \begin{pmatrix} \textcolor{rgb(118 139 240)}{d_{1,1}} & 0 & 0 & \dots & 0 \\ 0 & \textcolor{rgb(118 139 240)}{d_{2,2}} & 0 & \dots & 0 \\ 0 & 0 & \textcolor{rgb(118 139 240)}{d_{3,3}} & \dots & 0 \\ \hspace{0.1em}\vdots & \hspace{0.1em} \vdots & \hspace{0.1em} \vdots & \textcolor{rgb(118 139 240)}{\ddots} & \hspace{0.1em} \vdots \\ 0 & 0 & 0 & \dots & \textcolor{rgb(118 139 240)}{d_{n,n}} \end{pmatrix} $$
      $$ \forall (i, j) \in [\![1, n]\!]^2, \ (i \neq j) \Longrightarrow d_{i,j} = 0$$

      We also note the diagonal matrix \(D_n\) only in relation with its diagonal elements : \(D_n = diag(\lambda_1, \lambda_2, \ ..., \lambda_n)\).

   2. Identity matrix

      Identity matrix \(I_n\) are defined as follows:

      $$ I_n = \begin{pmatrix} \textcolor{rgb(118 139 240)}{1} & 0 & 0 & \dots & 0 \\ 0 & \textcolor{rgb(118 139 240)}{1} & 0 & \dots & 0 \\ 0 & 0 & \textcolor{rgb(118 139 240)}{1} & \dots & 0 \\ \vdots & \vdots & \vdots & \textcolor{rgb(118 139 240)}{\ddots} & \vdots \\ 0 & 0 & 0 & \dots & \textcolor{rgb(118 139 240)}{1} \\ \end{pmatrix} $$

      It is the square matrix of size \(n\) having the value \(1\) on its main diagonal, and \(0\) everywhere else. It's a specific case of diagonal matrix. For example,

      $$ I_3 = \begin{pmatrix} \textcolor{rgb(118 139 240)}{1} & 0 & 0 \\ 0 & \textcolor{rgb(118 139 240)}{1} & 0 \\ 0 & 0 & \textcolor{rgb(118 139 240)}{1} \end{pmatrix} $$
   3. Matrix of ones

      Matrix of ones \(J_n\) is the square matrix of size \(n\) where all elements are worth \(1\):

      $$ J_n = \begin{pmatrix} 1 & 1 & 1 & \dots & 1 \\ 1 & 1 & 1 & \dots & 1 \\ 1 & 1 & 1 & \dots & 1 \\ \vdots & \vdots & \vdots & \textcolor{rgb(118 139 240)}{\ddots} & \vdots \\ 1 & 1 & 1 & \dots & 1 \\ \end{pmatrix} $$

      So, for exemple \(J_3\) is worth:

      $$ J_3 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{pmatrix} $$

Matrix product

Associativity

Distributivity

Bilinearity

Multiplication by the identity

Diagonal product matrix

1. Product of two diagonal matrices
   $$ \forall \Bigl[ D_1 = diag(\lambda_1, \lambda_2, \ ..., \lambda_n), \ D_2 = diag(\mu_1, \mu_2, \ ..., \mu_n) \Bigr] \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K})^2, $$
   $$ D_1 \times D_2 = D_2 \times D_1 = diag \left(\lambda_1 \mu_1, \lambda_2 \mu_2, \ ..., \lambda_n \mu_n \right) $$
2. A diagonal matrix raised to the power of \(n\)
   $$ \forall \Bigl[ D = diag(\lambda_1, \lambda_2, \ ..., \lambda_n) \Bigr] \in \hspace{0.03em} \mathcal{M}_{n}(\mathbb{K}), $$
   $$ D^m = diag \left(\lambda_1^m, \lambda_2^m, \ ..., \lambda_n^m \right) $$

Matrix comatrix, transposed and determinant

Link between matrix, comatrix, transposed and determinant

Matrix transposition

Linearity of transposition

Transposed of a product

Inversion of matrix

Computation of l'inverse

Inverse of the inverse

Inverse of a transposed matrix

Inverse of a product

Both expressions \((9)\) and \((10)\) have the same behaviour:

So, the order of transposition or inversion has no importance,

Traces of matrix

Linearity of the trace

Trace of a product

Powers of matrices

Powers of the matrix of ones

Let \(J_n\) bet the matrix of ones of size \(n\).

Newton's binomial and Geometrical identity