Calculateur d'Inverse de Matrice 3x3

Entrez la Matrice A :
Visualisation de la Matrice

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Calculation Steps

Formulas:

1. Determinant of a 3x3 matrix:

\[|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

2. Adjoint matrix:

\[adj(A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \ C_{12} & C_{22} & C_{32} \ C_{13} & C_{23} & C_{33} \end{bmatrix}^T\]

Where \(C_{ij}\) is the cofactor of element \(a_{ij}\).

3. Inverse matrix:

\[A^{-1} = \frac{1}{|A|} adj(A)\]

Step-by-step calculation:

  1. Calculate the determinant: \[|A| = ( \cdot - \cdot ) - ( \cdot - \cdot ) - ( \cdot - \cdot ) = 0.00\]
  2. Calculate the cofactor matrix: \[C_{11} = (-1)^{1+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{12} = (-1)^{1+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{13} = (-1)^{1+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{21} = (-1)^{2+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{22} = (-1)^{2+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{23} = (-1)^{2+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{31} = (-1)^{3+1} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{32} = (-1)^{3+2} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]\[C_{33} = (-1)^{3+3} \cdot \begin{vmatrix} & \ & \end{vmatrix} = 0\]
  3. Transpose the cofactor matrix to get the adjoint: \[adj(A) = \begin{bmatrix}\end{bmatrix}\]

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Verification:

We can verify our result by multiplying A with its inverse (if it exists):


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Since the determinant is zero, the matrix is not invertible, and we cannot perform this verification.

Inverse, Déterminant et Adjointe de Matrice 3x3

Qu'est-ce qu'une Inverse de Matrice ?

L'inverse d'une matrice carrée A, notée A^(-1), est une matrice qui, lorsqu'elle est multipliée par A, donne la matrice identité. Pour une matrice 3x3 A, si A^(-1) existe :

\[A \cdot A^{-1} = A^{-1} \cdot A = I_3\]

Où I_3 est la matrice identité 3x3.

Formule de l'Inverse de Matrice

Pour une matrice 3x3 A, son inverse est donnée par :

\[A^{-1} = \frac{1}{|A|} \cdot adj(A)\]

Où : |A| est le déterminant de A adj(A) est l'adjointe (adjugate) de A

Déterminant d'une Matrice 3x3

Le déterminant d'une matrice 3x3 A = [a_ij] est calculé comme suit :

\[|A| = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})\]

Adjointe d'une Matrice 3x3

L'adjointe d'une matrice 3x3 A est la transposée de sa matrice des cofacteurs :

\[adj(A) = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{bmatrix}\]

Où C_ij est le cofacteur de l'élément a_ij.

Étapes de Calcul

  1. Calculer le déterminant de la matrice.
  2. Si le déterminant est non nul, continuer ; sinon, la matrice n'est pas inversible.
  3. Calculer la matrice des cofacteurs.
  4. Transposer la matrice des cofacteurs pour obtenir l'adjointe.
  5. Diviser l'adjointe par le déterminant pour obtenir l'inverse.

Exemple

Trouvons l'inverse de la matrice A :

\[A = \begin{bmatrix} 4 & 7 & 2 \\ 2 & 6 & 3 \\ 1 & 5 & 8 \end{bmatrix}\]

1. Calculer le déterminant :

\[|A| = 4(6 \cdot 8 - 3 \cdot 5) - 7(2 \cdot 8 - 3 \cdot 1) + 2(2 \cdot 5 - 6 \cdot 1) = 4(48 - 15) - 7(16 - 3) + 2(10 - 6) = 132 - 91 + 8 = 49\]

2. Calculer la matrice des cofacteurs :

\[C = \begin{bmatrix} 33 & -3 & -9 \\ -31 & 30 & -5 \\ 13 & -18 & 10 \end{bmatrix}\]

3. Transposer pour obtenir l'adjointe :

\[adj(A) = \begin{bmatrix} 33 & -31 & 13 \\ -3 & 30 & -18 \\ -9 & -5 & 10 \end{bmatrix}\]

4. Diviser par le déterminant pour obtenir l'inverse :

\[A^{-1} = \frac{1}{49} \begin{bmatrix} 33 & -31 & 13 \\ -3 & 30 & -18 \\ -9 & -5 & 10 \end{bmatrix} = \begin{bmatrix} \frac{33}{49} & -\frac{31}{49} & \frac{13}{49} \\ -\frac{3}{49} & \frac{30}{49} & -\frac{18}{49} \\ -\frac{9}{49} & -\frac{5}{49} & \frac{10}{49} \end{bmatrix}\]

Représentation Visuelle

A adj(A) A^(-1) 4 7 2 2 6 3 1 5 8 33 -31 13 -3 30 -18 -9 -5 10 0,67 -0,63 0,27 -0,06

La Formule de Multiplication de Matrices

Pour deux matrices 3x3 A et B, leur produit C = AB est défini comme :

\[C_{ij} = \sum_{k=1}^{3} A_{ik} \times B_{kj}\]

Où :

  • \(C_{ij}\) est l'élément dans la i-ème ligne et j-ème colonne de la matrice résultante C
  • \(A_{ik}\) est l'élément dans la i-ème ligne et k-ème colonne de la matrice A
  • \(B_{kj}\) est l'élément dans la k-ème ligne et j-ème colonne de la matrice B

Étapes de Calcul

Pour multiplier deux matrices 3x3 :

  1. Pour chaque élément de la matrice résultante :
    • Multiplier les éléments correspondants de la ligne de A et de la colonne de B
    • Additionner ces produits
  2. Répéter ce processus pour les 9 éléments de la matrice résultante

Exemple de Calcul

Multiplions la matrice A par la matrice B :

\[A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B = \begin{bmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}\]

Calcul du premier élément de C :

\[C_{11} = (1 \times 9) + (2 \times 6) + (3 \times 3) = 9 + 12 + 9 = 30\]

En continuant ce processus pour tous les éléments, nous obtenons :

\[C = AB = \begin{bmatrix} 30 & 24 & 18 \\ 84 & 69 & 54 \\ 138 & 114 & 90 \end{bmatrix}\]

Représentation Visuelle

A B A × B 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 30 24 18 84 69 54 138 114 90

Ce diagramme représente visuellement la multiplication de la matrice A par la matrice B, résultant en A × B.

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