Questions: Déterminer si les matrices suivantes sont idempotentes. a) A=[0 1; 0 1] b) B=[2 -1; 2 4] c) C=[-11 4; -33 12] d) E=[1 0 0; 0 1 1; 0 -1 1]

Déterminer si les matrices suivantes sont idempotentes. a) A=[0 1; 0 1] b) B=[2 -1; 2 4] c) C=[-11 4; -33 12] d) E=[1 0 0; 0 1 1; 0 -1 1]
Transcript text: 6. Déterminer si les matrices suivantes sont idempotentes. a) $A=\left[\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right]$ b) $B=\left[\begin{array}{cc}2 & -1 \\ 2 & 4\end{array}\right]$ c) $C=\left[\begin{array}{cc}-11 & 4 \\ -33 & 12\end{array}\right]$ d) $E=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1\end{array}\right]$
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Solution

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Solution Steps

To determine if a matrix is idempotent, we need to check if the matrix squared is equal to the matrix itself. In other words, for a matrix \( M \), we need to verify if \( M^2 = M \).

Solution Approach
  1. For each matrix, compute the square of the matrix.
  2. Compare the squared matrix with the original matrix.
  3. If they are equal, the matrix is idempotent; otherwise, it is not.
Step 1: Vérification de la matrice \( A \)

Pour la matrice \( A = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} \), nous calculons \( A^2 \): \[ A^2 = A \cdot A = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} = A \] Ainsi, \( A \) est idempotente.

Step 2: Vérification de la matrice \( B \)

Pour la matrice \( B = \begin{pmatrix} 2 & -1 \\ 2 & 4 \end{pmatrix} \), nous calculons \( B^2 \): \[ B^2 = B \cdot B = \begin{pmatrix} 2 & -1 \\ 2 & 4 \end{pmatrix} \cdot \begin{pmatrix} 2 & -1 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} 0 & -6 \\ 12 & 10 \end{pmatrix} \] Comme \( B^2 \neq B \), la matrice \( B \) n'est pas idempotente.

Step 3: Vérification de la matrice \( C \)

Pour la matrice \( C = \begin{pmatrix} -11 & 4 \\ -33 & 12 \end{pmatrix} \), nous calculons \( C^2 \): \[ C^2 = C \cdot C = \begin{pmatrix} -11 & 4 \\ -33 & 12 \end{pmatrix} \cdot \begin{pmatrix} -11 & 4 \\ -33 & 12 \end{pmatrix} = \begin{pmatrix} 121 - 132 & -44 + 48 \\ 363 - 396 & -132 + 144 \end{pmatrix} = \begin{pmatrix} -11 & 4 \\ -33 & 12 \end{pmatrix} = C \] Ainsi, \( C \) est idempotente.

Step 4: Vérification de la matrice \( E \)

Pour la matrice \( E = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{pmatrix} \), nous calculons \( E^2 \): \[ E^2 = E \cdot E = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & -1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 0 \end{pmatrix} \] Comme \( E^2 \neq E \), la matrice \( E \) n'est pas idempotente.

Final Answer

  • La matrice \( A \) est idempotente : \(\boxed{\text{Vrai}}\)
  • La matrice \( B \) n'est pas idempotente : \(\boxed{\text{Faux}}\)
  • La matrice \( C \) est idempotente : \(\boxed{\text{Vrai}}\)
  • La matrice \( E \) n'est pas idempotente : \(\boxed{\text{Faux}}\)
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