Mathxyz - upsc.xyz
0 votes
Let $A = \begin{bmatrix} 2 & 2 \\ 1 & 3 \end{bmatrix}$ . Find a non-singular matrix $P$ such that $P^{-1}AP$ is a diagonal matrix.
asked Nov 11, 2017 by randomisation (1,920 points)  

1 Answer

0 votes

Step 1: Calculate Eigenvalues

$ |A-\lambda I| = 0 \implies (2-\lambda)(3-\lambda) - 2 = 0 \implies \lambda = 1, 4 $

 

Step 2: Calculate Eigenvectors corresponding to the eigenvalues

For $\lambda = 1, (A-I)X = 0 \implies \begin{bmatrix} 1 & 2 \\ 1 & 2 \end{bmatrix} \begin{bmatrix}  x_1 \\ x_2 \end{bmatrix} \\ \implies x_1 + 2x_2 = 0. \\ For x_2 = 1, x_1 = -2 \implies \begin{bmatrix}  x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix}  -2 \\ 1 \end{bmatrix} $

 

For $\lambda = 4, (A-I)X = 0 \implies \begin{bmatrix} -2 & 2 \\ 1 & -1 \end{bmatrix} \begin{bmatrix}  x_1 \\ x_2 \end{bmatrix} \\ \implies x_1 - x_2 = 0. \\ For x_2 = 1, x_1 = 1 \implies \begin{bmatrix}  x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix}  1 \\ 1 \end{bmatrix} $

 

Step 3: Form the matrix

The required non-singular matrix $P$ such that $P^{-1}AP$ is a diagonal matrix is the matrix with the eigenvectors as its columns. Hence, $P = \begin{bmatrix} -2 & 1 \\ 1 & 1 \end{bmatrix}$.

answered Nov 11, 2017 by randomisation (1,920 points)  
Welcome to MathXyz, where you can ask questions and receive answers from other members of the community. Please strictly ask questions from UPSC Mathematics syllabus.
36 questions
26 answers
2 comments
12 users