Question

Diagonalise the matrix 𝐴 = [ 𝑎 𝑏 𝑐 𝑑 ], where 𝑎 + 𝑐 = 𝑏 + 𝑑, by finding a nonsingular matrix 𝑃 and a diagonal matrix 𝐷 such that 𝐴 = 𝑃𝐷𝑃 −1 . 

Answer 1

$\text{let A =}\begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix}\\ \text{to get the eigen values, }\\ t^2-tr(A)t-|A|\\ t^2-4t-5=(t+1)(t-5)\\ \lambda_1=-1, \lambda_2=5\\ \text{to compose the eigen vectors,}\\ \begin{bmatrix} 3-(-1) & 4 \\ 2 & 1-(-1) \end{bmatrix}\\ =\begin{bmatrix} 4 & 4 \\ 2 & 2 \end{bmatrix}\\ \text{corresponding to} \\ 4x+4y=0\\ \text{the eigen vector is}\\ v_1=\begin{vmatrix} -1\\ 1 \end{vmatrix}\\ \text{also, }\\ \begin{bmatrix} 3-(5) & 4 \\ 2 & 1-(5) \end{bmatrix}\\ =\begin{bmatrix} -2 & 4 \\ 2 & -4 \end{bmatrix}\\ \text{corresponding to} \\ 2x-4y=0, x-2y=0\\ \text{the eigen vector is}\\ v_2=\begin{vmatrix} 2\\ 1 \end{vmatrix}\\ \text{then p}= \begin{bmatrix} -1 & 2 \\ 1 & 1 \end{bmatrix}\\ p^{-1}=\frac{1}{-3} \begin{bmatrix} 1 & -2\\ -1 & -1 \end{bmatrix}\\ \text{setting D to be a diagonal matrix}\\ D=\begin{bmatrix} -1 & 0 \\ 0 & 5 \end{bmatrix}\\ A=PDP^{-1}$