IGNOU MTE 2 2018 - Check whether the matrices A and B are diagonalisable. Diagonalise tho

Question

Check whether the matrices A and B are diagonalisable. Diagonalise those matrices which are diagonalisable.

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Image ignouacademy-com--p-diagonalise-49919

02 Feb 2018

Answer :

Word Count : 503

To determine whether the given matrices \( A \) and \( B \) are diagonalizable, we need to check if they have enough linearly independent eigenvectors to form a basis for the space. If they do, we can diagonalize them by expressing them in the form \( P D P^{-1} \), where \( D \) is a diagonal matrix of eigenvalues, and \( P \) is a matrix of corresponding eigenvectors. ### Step 1: Identify the Matrices First, let's extract the matrices \( A \) and \( B \) from the provided images. #### Matrix \( A \): \[ A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \] #### Matrix \( B \): \[ B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \] ### Step 2: Check Diagonalizability of Matrix \( A \) To check if \( A \) is diagonalizable, we need to find its eigenvalues and eigenvectors. #### Eigenvalues of \( A \): The eigenvalues are found by solving the characteristic equation: \[ \det(A - \lambda I) = 0 \] \[ \det\begin{pmatrix} 1

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